Finite Minds

Here, I explain what’s wrong with infinitism in epistemology: the human mind is finite.*

[ *Based on: “Finite Minds,” pp. 171-87 in Themes from Klein: Knowledge, Scepticism, and Justification, ed. Cherie Braden, Branden Fitelson, and Rodrigo Borges (Springer, 2019). This was a volume created in honor of epistemologist Peter Klein, my PhD advisor. Another student of his, Cherie Braden, was the main impetus for the project. ]

1. Background: About Infinitism

Some people think that to know P, you need a reason for believing P, and the reason must also be something you know. This threatens to create an infinite regress. There are 4 reactions to this in epistemology:

1. Foundationalism: Reject the starting assumption. There must be things you can know that don’t depend on reasons (or there are things that can serve as reasons without being known).

2. Skepticism: We can’t know anything.

3. Coherentism: Maybe something like circular reasoning is okay.

4. Infinitism: You can have an infinite regress, and that’s okay.

Infinitism is historically the least popular view, defended, as far as I know, by no one prior to Peter Klein.

To make it seem less crazy and hence more worth discussing: Infinitists do not propose that anyone actually completes an infinite chain of reasoning. They only propose a potential infinity: a belief is justified only if the believer could provide a reason for it if it were challenged, and then could provide a reason for that reason if it were challenged, and so on.

Klein also suggests that, as you go through the process of providing such reasons, your belief becomes more justified. It never becomes perfectly justified, though, because you never complete an entire infinite series.

The Finite Mind Objection: The human mind is finite, and therefore no one can have an infinite series of reasons available to them.

Klein’s reply: Initially, he suggested that a person can have infinitely many dispositional beliefs, even though you can’t have infinitely many occurrent beliefs. (Background:

• Occurrent beliefs: Beliefs that you are thinking right now.

• Dispositional beliefs: Things that you believe but are not thinking of at the moment. Ex.: say you have a neighbor who is a Mormon. I ask you about his beliefs. You can say, “My neighbor believes Joseph Smith was a prophet” — even though, in all probability, your neighbor is not thinking about that proposition at the moment. He believes it dispositionally.

  You might have infinitely many dispositional beliefs, e.g., that 2>1, 3>1, etc.)

Later, Klein suggested that it might be enough for a person to merely have some belief-forming practice with the potential to generate each of infinitely many beliefs.

Now I’m going to defend the Finite Mind Objection.

2. When Are Reasons Available?

Again, the question is whether a person can have infinitely many available reasons. Klein suggests that P might be available to you as a reason if you dispositionally believe P, or even if you have a belief-forming practice that would lead you to endorse P.

2.1. Dispositional Beliefs

Not just anything that you could or would come to believe counts as something that you (currently) dispositionally believe. I think there are three ways of counting as dispositionally believing P:

1. Having occurrently believed P at some earlier time, without having changed one’s mind or forgotten it.

2. Having an appearance that P with the disposition to take the appearance at face value.

3. Having a belief (or other mental state) representing that Q, where Q (fairly) directly and obviously entails P.

Note the “directly and obviously” clause. Example: in the Meno, Socrates leads a slave boy along a chain of reasoning, using geometric figures, to the conclusion that the diagonal of a square is Sqrt(2)*(the length of the side). Socrates doesn’t actually tell the slave any of the premises or steps of the reasoning; rather, he does it entirely by asking questions whose answers are obvious to the slave. At the end, Socrates claims that this shows that the slave already knew that the diagonal is (sqrt(2)*the side). I hope you agree with me that that’s false. The slave did not in any sense know that (dispositionally or otherwise); Socrates caused him to acquire the knowledge.

This severely limits how many dispositional beliefs you can count as having. More on that later.

2.2. Belief-forming Practices

For P to be an “available” reason, I think it’s not enough to have a belief-forming practice that would lead you to accept P.

Example: Sue believes that there are unicorns, just because she whimsically made up that idea. Suppose that if someone were to challenge her belief, she would start looking for unicorns, and she would start by looking under her bed. As chance would have it, there is actually a baby unicorn under her bed, so she would then see it and use that to justify her belief that unicorns exist. However, no ever actually challenges her belief, so she never bothers to look for unicorns and never in fact sees any (nor does she ever have any other evidence for them). End of story.

Intuitively, Sue’s belief in unicorns is arbitrary and not at all justified. This shows that, to the extent that “available reasons” are supposed to explain why beliefs count as justified and not arbitrary, it is not enough to merely have a practice that would lead to your acquiring a given piece of information; that does not suffice for that information to count as presently “available”.

This, again, severely limits how many reasons can be available to a person at a given time.

3. Examples of Infinite Belief Systems

The issue here is whether there is any way in which a human being (given our finite minds) could have infinitely many beliefs or available propositions. Let’s look at some potential examples:

3.1. Math

Suggestion: Maybe you have infinitely many (dispositional) mathematical beliefs: e.g., you believe 2>1, 3>1, 4>1, etc.

Problem: No, because there is some point in the series at which you would be unable to think about the numbers. Some numbers are simply too large for a normal human to comprehend them or even refer to them. Perhaps the best way to see this point is to read a little about Graham’s Number. This is a ridiculously large integer, far too large to write in decimal notation, or even to approximate using scientific notation, using all the resources of the known universe. It takes several minutes to understand the explanation of what this number is (so I won’t try to explain it here), and many people in fact could not understand it at all. Even if you can comprehend Graham’s Number, there is a large number of other numbers, in between, say, Graham’s Number and 2*(Graham’s Number), such that you have no way of referring to them individually. About those numbers you do not have any beliefs. So you do not in fact believe all the propositions in the series, {[2>1], [3>1], [4>1], …}.

Suggestion: No, you really do have the capacity to refer to all those numbers in principle, since you could read a really long tape with a printout of the number in decimal notation. This would require you to also have a really long lifespan. But the facts that your lifespan is limited and there aren’t enough particles in the universe to create that tape are mere “in practice” limitations that are irrelevant to your “in principle” capabilities.

Problem: But surely your actual cognitive limitations are not irrelevant in this conversation. And even if there were such a tape and you were immortal, you still could not grasp the number on that tape, since, 500 years into reading it (or sooner), you would have forgotten what the beginning of the tape said. At no point could you hold the whole number in your mind.

3.2. Colors

Suggestion: Suppose I see the sky to be some particular shade of blue. I believe that it is that exact shade. I thus count as dispositionally believing, of every other shade, that the sky isn’t that other shade. Since color varies continuously, this is an infinite set of dispositional beliefs.

Problem: No, because the human eye has limited discriminatory capacity. (Experiments suggest that we can discriminate no more than 2 million colors in ideal conditions, and of course we have far fewer color concepts than that.) So, for shades that are sufficiently similar to each other, I cannot have distinct beliefs about those shades.

3.3. Indexicals

Suggestion: Suppose there is an infinitely long red carpet. I could look at any given small part of the carpet and believe [that is red]. (The word “that” is an “indexical” term.) So I have the capacity for infinitely many indexical beliefs.

Problem: Human beings have limited discriminatory capacities in general. So for pairs of objects that are sufficiently similar (2 portions of carpet, or 2 shades of color, etc.), I would be unable to tell that they were distinct. I could still refer to such objects separately by saying “that object” on two distinct occasions. However, I would be unable to tell that I was referring to distinct objects. For this reason, my beliefs about the objects could not rationally play distinct roles in my reasoning, since I could not tell that they were distinct beliefs. E.g., when I think [that is red] and [that is red] on two distinct occasions, I could not rationally treat those as two distinct reasons since I could not tell that they were distinct propositions.

Problem: At any given time, a human being can only have perceived finitely many objects. Facts about objects that you merely could perceive but haven’t are not “available” to you in the relevant sense (see 2.2 above). Furthermore, even if you were immortal, you could only remember so many objects. So you can only have finitely many indexical propositions available at a given time.

4. Entertainable Propositions

More generally, I think there are only finitely many propositions that a given person could in principle entertain. This is because:

• There are only finitely many things that a person could directly refer to (see above points about limited discriminatory capacity & limited memory; we also have only finitely many concepts, and there is a limit to how complex a concept that we can grasp may be).

• There are only finitely many basic predicates that we can understand and distinguish. Again, this is due to finite discriminatory capacities and conceptual abilities.

• There is a limit to the propositional structures that we can grasp. Granted, a finite number of terms and predicates can give rise to an unlimited set of propositions. But this would be a set of more and more complex propositions, and above a certain degree of complexity, we can no longer grasp the proposition. (E.g., there is [Fa], [Fa v Fb], [Fa → (Fa v Fb)], [Fa v (Fa → (Fa v Fb))], etc.)

• Similarly, there is a limit to the complexity of the predicates that you could grasp based on your finite set of basic predicates and terms.

Btw, notice that human beings periodically come up with new concepts not previously grasped in human history, e.g., ‘hyperreal numbers’, ‘stock derivatives’, or ‘MRI scans’. For most of human history, no one entertained any propositions at all about those things, but now we do. So the class of propositions people can think about keeps expanding.

But notice also that these new concepts are increasingly abstract, complex, or in general increasingly hard to grasp. Indeed, there are some people today who could not understand any of those things that I just mentioned, even if you tried to explain them. So, given our finite intellects, we can infer that there is a limit to how long this can go on. While there might in some sense exist infinitely many abstract objects in Plato’s heaven, at some point even the smartest humans would be unable to grasp them.

5. Indeterminacy

How many reasons do I have available to me? The answer to that is probably indeterminate for various reasons. There is the vagueness of “availability”, there is indeterminacy about my discriminatory capacities, there is indeterminacy about how many propositions I am aware of, etc. (As you survey the scene in front of you right now, exactly how many propositions do you thereby see to be correct?)

However, this isn’t a problem. My argument doesn’t require that there be a specific finite number that is determinately the number of reasons you have available. It just requires that the number is definitely not “infinity”. Which it pretty clearly isn’t.