The Man Who Understood Infinity

[Catherine Nicolo]

Forexample, at least, we can take into account integers. We can think, principally, and "understand" infinitely many numbers that are displayed on the screen. Nowadays, we are trying to design artificial intelligence which is capable at least human being. However, I am stuck with infinity. I try to find a way how can teach a model (deep or not) to understand infinity. I define "understanding' in a functional approach. For example, If a computer can differentiate 10 different numbers or things, it means that it really understand these different things somehow. This is the basic straight forward approach to "understanding".

As I mentioned before, humans understand infinity because they are capable, at least, counting infinite integers, in principle. From this point of view, if I want to create a model, the model is actually a function in an abstract sense, this model must differentiate infinitely many numbers. Since computers are digital machines which have limited capacity to model such an infinite function, how can I create a model that differentiates infinitely many integers?

For example, we can take a deep learning vision model that recognizes numbers on the card. This model must assign a number to each different card to differentiate each integer. Since there exist infinite numbers of integer, how can the model assign different number to each integer, like a human being, on the digital computers? If it cannot differentiate infinite things, how does it understand infinity?

Let's suppose that there's something special about infinity (or about continuous concepts) that makes them especially difficult for AI. For this to be true, it must both be the case that humans can understand these concepts while they remain alien to machines, and that there exist other concepts that are not like infinity that both humans and machines can understand. What I'm going to show in this answer is that wanting both of these things leads to a contradiction.

If by understanding, we mean that a computer has the conscious experience of a concept, then we quickly become trapped in metaphysics. There is a long running, and essentially open debate about whether computers can "understand" anything in this sense, and even at times, about whether humans can! You might as well ask whether a computer can "understand" that 2+2=4. Therefore, if there's something special about understanding infinity, it cannot be related to "understanding" in the sense of subjective experience.

So, let's suppose that by "understand", we have some more specific definition in mind. Something that would make a concept like infinity more complicated for a computer to "understand" than a concept like arithmetic. Our more concrete definition for "understanding" must relate to some objectively measurable capacity or ability related to the concept (otherwise, we're back in the land of subjective experience). Let's consider what capacity or ability might we pick that would make infinity a special concept, understood by humans and not machines, unlike say, arithmetic.

We might say that a computer (or a person) understands a concept if it can provide a correct definition of that concept. However, if even one human understands infinity by this definition, then it should be easy for them to write down the definition. Once the definition is written down, a computer program can output it. Now the computer "understands" infinity too. This definition doesn't work for our purposes.

We might say that an entity understands a concept if it can apply the concept correctly. Again, if even the one person understands how to apply the concept of infinity correctly, then we only need to record the rules they are using to reason about the concept, and we can write a program that reproduces the behavior of this system of rules. Infinity is actually very well characterized as a concept, captured in ideas like Aleph Numbers. It is not impractical to encode these systems of rules in a computer, at least up to the level that any human understands them. Therefore, computers can "understand" infinity up to the same level of understanding as humans by this definition as well. So this definition doesn't work for our purposes.

We might say that an entity "understands" a concept if it can logically relate that concept to arbitrary new ideas. This is probably the strongest definition, but we would need to be pretty careful here: very few humans (proportionately) have a deep understanding of a concept like infinity. Even fewer can readily relate it to arbitrary new concepts. Further, algorithms like the General Problem Solver can, in principal, derive any logical consequences from a given body of facts, given enough time. Perhaps under this definition computers understand infinity better than most humans, and there is certainly no reason to suppose that our existing algorithms will not further improve this capability over time. This definition does not seem to meet our requirements either.

Finally, we might say that an entity "understands" a concept if it can generate examples of it. For example, I can generate examples of problems in arithmetic, and their solutions. Under this definition, I probably do not "understand" infinity, because I cannot actually point to or create any concrete thing in the real world that is definitely infinite. I cannot, for instance, actually write down an infinitely long list of numbers, merely formulas that express ways to create ever longer lists by investing ever more effort in writing them out. A computer ought to be at least as good as me at this. This definition also does not work.

This is not an exhaustive list of possible definitions of "understands", but we have covered "understands" as I understand it pretty well. Under every definition of understanding, there isn't anything special about infinity that separates it from other mathematical concepts.

So the upshot is that, either you decide a computer doesn't "understand" anything at all, or there's no particularly good reason to suppose that infinity is harder to understand than other logical concepts. If you disagree, you need to provide a concrete definition of "understanding" that does separate understanding of infinity from other concepts, and that doesn't depend on subjective experiences (unless you want to claim your particular metaphysical views are universally correct, but that's a hard argument to make).

Infinity has a sort of semi-mystical status among the lay public, but it's really just like any other mathematical system of rules: if we can write down the rules by which infinity operates, a computer can do them as well as a human can (or better).

You seem to assume that to "understand"(*) infinities requires infinite processing capacity, and imply that humans have just that, since you present them as the opposite to limited, finite computers.

But humans also have finite processing capacity. We are beings built of a finite number of elementary particles, forming a finite number of atoms, forming a finite number of nerve cells. If we can, in one way or another, "understand" infinities, then surely finite computers can also be built that can.

While humans can, e.g. use infinities when calculating limits etc. and can think of the idea of something getting arbitrarily larger, we can only do it in the abstract, not in the sense being able to process arbitrarily large numbers. The same rules we use for mathematics could also be taught to a computer.

TL;DR: The subtleties of infinity are made apparent in the notion of unboundedness. Unboundedness is finitely definable. "Infinite things" are really things with unbounded natures. Infinity is best understood not as a thing but as a concept. Humans theoretically possess unbounded abilities not infinite abilities (eg to count to any arbitrary number as opposed to "counting to infinity"). A machine can be made to recognize unboundedness.

Our brains are not infinite (lest you believe in some metaphysics). So, we do not "think infinity". Thus, what we purport as infinity is best understood as some finite mental concept against which we can "compare" other concepts.

Our concept of quantity/number is unbounded. That is, for any any finite value we have a finite/concrete way or producing another value which is strictly larger/smaller. That is, Provided finite time we could only count finite amounts.

You cannot be "given infinite time" to "count all the numbers" this would imply a "finishing" which directly contradicts the notion of infinity. Unless you believe humans have metaphysical properties which allow them to "consistently" embody a paradox. Additionally how would you answer: What was the last number you counted? With no "last number" there is never a "finish" and hence never an "end" to your counting. That is you can never "have enough" time/resources to "count to infinity."

However, what we are really doing is: Within our bounds we are talking about our bounds and, when ever we need to, we can expand our bounds (by a finite amount). And we can even talk about the nature of expanding our bounds. Thus:

A process/thing/idea/object is deemed unbounded if given some measure of its quantity/volume/existence we can in a finite way produce an "extension" of that object which has a measure we deem "larger" (or "smaller" in the case of infinitesimals) than the previous measure and that this extension process can be applied to the nascent object (ie the process is recursive).

in- :a prefix of Latin origin, corresponding to English un-, having a negative or privative force, freely used as an English formative, especially of adjectives and their derivatives and of nouns (inattention; indefensible; inexpensive; inorganic; invariable). (source)

So in-finity is really un-finity which is not having limits or bounds. But we can be more precise here because we can all agree the natural numbers are infinite but any given natural number is finite. So what gives? Simple: the natural numbers satisfy our unboundedness criterium and thus we say "the natural numbers are infinite."

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