Monday, January 17, 2005

Infinite Knowledge

A post over at Scottish Nous got me wondering whether I know infinitely many things.

I think perhaps I do. For I know all of the following:
• 1 is a number

• 2 is a number

• 3 is a number

• . . .

• 1 000 000 000 is a number

There are infinitely many statements in that list, and I know every one of them, so it seems like I must have an infinite amount of knowledge. Yay for me!

Is there any problem with this reasoning? Being a bear of very little (certainly finite) brain, I presumably don't have all that knowledge actually tokened 'inside' me - how could it possibly fit? It must be merely potential, I think, and is perhaps best explained in dispositional terms: If someone were to ask me 'is X a number?', for any particular X (and there are infinitely many options they could choose from), I could answer it correctly.

I've touched on some similar issues before. I'd just like to get sampling of others' opinions here: are you happy to grant that we have infinite knowledge? Or would you rather restrict knowledge to actually-tokened thoughts, to the exclusion of 'potential' ones that we're merely disposed towards affirming?

1. Depends on how you define knowledge. You don't know each number as a fact, but you can derive any of them. You could reasonably define knowledge either way, I expect.

Big infinity, small infinity. It's all relative. There are plenty of bigger sets around than the natural numbers, it's not like you know everything.

Cheers,
-MP

p.s. would be interested in having you respond again over on MP.com, on the beliefs/facts post.

Posted by Tennessee Leeuwenburg

2. Interesting question. What seems obvious (to me) is that you don't have an infinite number of *beliefs*, since, as I understand the term, a belief must be tokened "inside" you in order to count as a belief. So, if your knowledge is simply a subset of your beliefs, as seems plausible to assume, then I say: No, you don't have an infinite amount of knowledge.

Posted by david

3. If you are interested in this topic, you might want to check out Smyth's Reading Peirce Reading which actually has a rather involved discussion of this via the connection between neoPlatonic conceptions of knowledge and Peirce's conception of knowledge. Way back in the early days of my blog I summarized his assertions.

Posted by Clark

4. Actually, I don't think you really don't know that much. In an algorithmic information sense, you don't know very much at all since what you know can be compressed into:

1 is a number
if n is a number, so is n+1

And as you said, you haven't "tokened" every single n.

You said that you could answer 'is X a number?' for any single number. But that doesn't imply you know infinitely many numbers, it implies (at most) that you could run the formula I gave above until you got to (tokened) any number. But in order to token an infinite amount of numbers, you would need an infinite amount of time. You would have to run that fomula ad infinitum. You can't ever "get to" infinity. You could keep answering the "is X a number" question until you die and you still wouldn't know an infinite number of things.

Again, in an algorithm information sense, I think the only way you can say that you know an infinite number of things is if you can in some way compress that infinite into finite terms. This could be through induction or the "..." symbol or the aleph-0 symbol or any other kind of trick we can think of.

If knowledge must be tokened, then you can't have infinite knowledge. If knowledge merely means "able to be tokened" then you can have infinite knowledge.

Posted by Macht

5. David is right to mention beliefs -- infinite knowledge will rise and fall with infinite beliefs, since a piece of knowledge is just a certain kind of belief.

I don't share David's intuition, though, that beliefs must be tokened. I believe that you, Richard, have hands. I even had that belief before I started writing this comment, at a point in which that question had never even occurred to me.

I have no problem with beliefs as dispositions. Likewise, then, with knowledge. Some people fuss about using the terms that way. I don't really want to argue about definitions, but it seems like the dispositional things that I want to call beliefs are similar to uncontroversial beliefs in many important ways.

Posted by Jonathan

6. "it's not like you know everything"

Shush, you, don't spoil my fun :)

"as I understand the term, a belief must be tokened "inside" you in order to count as a belief"

Yeah, that's equivalent to the question I'm getting at here. I'm not convinced that's the right answer though. If I asked, "do you believe that elephants are larger than earwigs?", you'd likely answer something like "yes, of course!", even if you've never entertained that exact thought before. I next ask: "Did you also believe that yesterday, or did you only just start to believe it after I asked my original question?" I think it would be most plausible to answer that you've believed it for a long time, despite never 'tokening' the thought until now.

In short, I think I'd prefer to say that a belief is a dispositional state, rather than requiring an actual 'tokening' of a thought.

Clark - thanks for the link, I'll have to check it out in more detail when I have time...

Macht - I certainly can't provide all of those infinite answers. I meant instead to suggest that I could answer any one of those infinitely many possible questions. I took that to imply that I knew them all (according to one - I think plausible - interpretation of 'know'). But your last paragraph sums up the problem I'm asking about: Does knowledge require tokening?

Posted by Richard

7. Oh, I see Jonathan bet me to it. Let me add, then, that I agree with everything he just wrote!

Posted by Richard

8. My own sense is that "calculations" are never required in order to access one's beliefs; one need only remember one's beliefs in order to access them. So I would say that five minutes ago, I did not believe that elephants are larger than earwigs, or that Richard had hands. (Of course, I didn't doubt those propositions either; I just didn't have any beliefs at all about them.)

Moreover, even if others don't share my intuitions here, I think it can be shown that the notion of "dispositional" beliefs would produce an unworkable notion of belief. If someone were to ask me whether X is a number, I could give an answer. But this is not an adequate test for belief. For if someone were to ask me whether X is equal to the sum of Y and Z, I could give an answer to that question as well. But of course, prior to considering the question, I had no beliefs about its answer. There does not seem to me any principled way to draw a line between the question "Is X a number?" and "Is X the sum of Y and Z?" The only relevant difference between them that I see is that the calculation required to answer the former question requires less effort than that required to answer the latter question. So if we allow dispositional belief about the answer to the former question, I suppose we must allow dispositional belief about the answer to the latter question. But that would be absurd.

One final point: If we do end up allowing dispositional knowledge to exist, then we'll no longer have to choose between foundationalism and coherentism. There is a third alternative, which I call "linearism." A linearist chain of inference is infinitely long and non-circular. Such a chain involves an infinite number of inferentially-linked beliefs, strung in sequence. Normally, I think linearism is absurd because it requires a subject to have an infinite number of beliefs. But this "dispositional" construal of knowledge and belief makes it possible to have an infinite number of beliefs. So linearism is not absurd -- at least, not for the reason I would normally give.

Sorry for the too-long comment!

Posted by david

9. Does the question "Is X a number?" have an infinite number of ways it can actually be asked? The person asking the question would have to token X in order to ask the question. If so, he may not be able to ask any of an infinite number of questions and, thus, you may not be able to answer an infinite number of them. So I don't know if this would be an example of infinite knowledge even if you were able to give answers to all the questions that could be tokened.

Posted by Macht

10. I agree that it's fine to define belief so that you can believe X even when you have never tokened X. However, defining belief in terms of dispositions to token is tricky, because it is not the case that you are capable of tokening any one of the infinitely many statements on that list. Even if you were capable of tokening up to 10 digits per second and you spent every second of the next 100 years tokening the digits of a single number, then you would not be able to token any numbers of more than 32 billion digits.

You don't have the potential to token a belief about numbers larger than 10^32,000,000,000 . If we accept that you believed that 7486193418 is a number even before you just tokened it, though, then I think that we'd also want to claim that, for each number X between 10^32,000,000,000 and 10^32,000,000,001 , you believe that X is a number. I'm not sure how such a concept of belief could be made precise.

Posted by Blar

11. This question is very well covered in Braithwaite's Introduction of the Dover translation of Godel's, "On Formally Undecidable Propositions...". (for a quick lookup. Godel's article, "What is Cantor's continuum problem?" (1947) addresses the specific argument).

Speaking Informally... I think there's some quick ways to clarify what I believe to be the main question. There is a distinct difference between knowing a combination of [0,1,2,3,4,5,6,7,8,9], [Lx, Lxx, Lxxx, Lxxxx,... Lxn+1], and ['x' is a Number], can be used to derive a set with an infinite membership, and between believing this demonstrates an 'infinite amount of knowledge'. Obviously a finite set of symbols, definitions and rules ('knowledge' type A) are not the same as an infinite set of data ('knowledge' type B).

Put simply, "the [knowledge] OF a [set] WITH [infinite membership]" is not: "a [set] OF [knowledge] WITH [infinite membership]". Here's the link to a Berkeley site (http://www.icsi.berkeley.edu/~kay/bcg/ConGram.html).

"Potential" or "possible" or "hypothetical" things don't typically have enough existential import to merit trying to reason out. This includes Universes of Discourse that encompass, 'all possible worlds', which is a cheap and unphilosophical gimmick 99.99999% of the times it's used. heh.

Posted by A. Scott Crawford

12. knowledge is something we may have a grasp at but think of the word "infinite" it is endless! we can only percieve what comes in thru our vision and auditory senses., infinite is reserved for the divine if you so choose to believe (opinion wise) numbers,formulas, "Xs' AND "v" yes are equations to formulas of scientific entries that i only wish i knew! as eienstien did, should we be so blessed to comprehend according to post atomic theories! in a nutshell humans have a limited amount of "knowledge" more than the computer yes! but to figure out how to end the timer of age? to live beyond 80,100,120? the infinite is just that! without end! alpha and omega!as with "DIO" ITS RESERVED FOR THOSE THAT ARE OF ANOTHER DIMENSION.

13. Hi guys

I found you by accident, as philosophy isn’t my thing (as you will notice) by your topic caught my eye and I could resist… don’t worry I won’t make a habit of it.

Most of what I have to say is probably 101 stuff to you guys but your topic caused me to pause and think for a while and for that I thank you all but please remember my ravings are that of a layman… so be gentle.

What exactly is knowledge? From my limited understand of the subject I would suggest that knowledge is a generic label / term used to describe a repository of data and information accumulated from our observations of the processes that form the environment we perceive around us.

How is knowledge used? I would content that knowledge is used by that element of our individual entity, referred to as our consciousness, which in itself must contain a repository of Meta data, used to contextualise the data and information stored in the knowledge repository, thus converting the data/ information into representations that have perceived meaning to us.

So getting back to the question ‘Do I have infinite knowledge?’ I think that to be able to answer this question it is necessary to first of all question the question, in an attempt to understand exactly what is being asked.

Is the question asking whether I can access a repository of data / information, which contains data / information on all things in existence or ever will exist? If so then the question seems to be asking if our consciousness can exist at every point in time and space at the same instant.

Or is the question asking whether my stored Meta data can process data / information available to me and arrive at a correct meaning to it. If this is the case then the question isn’t really asking if I have infinite knowledge, rather it is asking whether I have the stored Mata data (or Meta functions) to understand all things.

If the question has the former meaning then I would have to respectfully suggest that none of us are yet ready for god hood. If the latter meaning is meant then I would argue, that some humans already posses these abilities in a limited context, more normally referred to as a gift.

14. you guys have it wrong (maybe not for everyone, but probably for the average joe). A moron could tell you that 123890239058290238398 is a number, but he might not know what the formula is for generating the infinite series...

1 is a number, 2 is a number,..... 100000 is a number - you dont know that 29038509238309 is a number, what you know is this:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are numbers and any combination of those numbers is a number.

-dale`

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