Can Brain in A Vat learn Math?

Recently, I’ve been thinking about the perpetual illusion scenario which happens by mere chance. I will give a longish presentation for those who missed previous posts on this, and then shortish idea of a direction to think in.


The scenario is your usual brain in a vat / Matrix scenario with one twist – our sensory organs are connected to random generators, so whole illusion happens by mere chance. The twist is there in order to remove any possibility for representational or other indirect causal relation to relevant things which would give rise to the concepts that the BiV might become aware of.

It seems unproblematic that the BiV in such situation, let’s call him Neo, hasn’t become aware of any particular thing in the world, nor he has become aware of any natural kind. But, it seems, that there is no reason to deny that Neo has become aware of the notion of thing in general, the notion of color (in general, and even specific colors in particular), size, form; then notion of causality, change in general (and specific kinds of change in particular, e.g. movement); notion of awareness, notion of other subjects (what some call “other minds”), further – notion of social practices, communication, artifacts and so on…

The reasoning behind this my conviction would go something like this – I can’t see any logical impossibility of connecting a person to such random inputs, in such way that the body is affected in same way as it is affected in reality. Further, I don’t see any logical impossibility for this set of inputs to be generated by a random generator. True, the chance of something like that happening might be 1:gajillion on kajillionth power, but still I don’t see reason to think it is impossible. Now, if we imagine that Neo’s body has had by chance the same ‘inputs’ like mine (also given that we have sufficiently similar bodies), I don’t see a reason why we would assume that he is not having any kind of thought.

But now, if we disconnect Neo and return him into the real world, everything seems OK! We can communicate with him, discuss math problems, discuss social issues, he acts normally (given that his muscles somehow were kept in shape while he was in the vat), and so on.

Even he would use the words for natural kinds normally (and even proper names), I buy the point that his words didn’t mean those things while he was in the vat. They couldn’t mean those things, because he wasn’t aware of those things at all. This is true in same way as a fictional character is still fictional even by mere chance its description and story fits some person exactly. It seems to me that our intuition fights against this conclusion, just because that our thinking because of implausibility right away assumes that there must be some connection between those, but the moment we assume that connection, we have reasons to think that the story is about that person. If somebody has read a story about the fictional person, he can’t say that he knows the real person if somebody doesn’t tell him that “by strange coincidence all the facts about fictional person are true about this real person”. I think the same happens to our intuition in case of Neo’s use of proper names and names of natural kinds – it is our awareness of the fact that everything coincides that pushes our intuition to think that there is a relation between the two. But as posited there are no relation, and Neo can only know that the illusionary facts can be applied to real e.g. rabbits, only if he knows what we do – that by mere chance he got the same inputs like mine. Only with that knowledge, he can feel justified to apply his “knowledge” of illusionary rabbits to real rabbits.

But, the situation seems different to me for those abstract notions I mentioned. I don’t see reason to think that Neo didn’t became aware of Pythagorean theorem while in the vat. To make analogy with the fiction – even some theorem is presented in the fictional story with a proof and everything, it is not as if it is a ‘fictional theorem’. And the same it seems to me goes for all those other abstract notions I named, starting from thing, multitude of things, etc… There is no need for him to somehow ‘transfer’ the fictional facts to some new entities. Those notions are the same ones.

But, then, given that you agree with my reasoning here – the issue appears – how is it possible that Neo has become aware of those things, when he wasn’t at any time acquainted , nor causally related in any relevant way to those things?


So, this has been what has been puzzling me.

One approach is to take those notions as innate in some way. Some form of Kantianism I guess – take those notions as pure concepts of understanding, something that is there prior to any experience. In such a picture it is no wonder that Neo will became aware of them even given the weird events in which he participates. It sounds good, but some of us are not Kantians.

The first idea that came to my mind, and I have already mentioned this, is that Neo became aware of possibilities. For example he didn’t become aware of (any) things, but he became aware of possibility for there to be things. He didn’t became aware of multitudes, but of possibilities for there to be multitudes (couples, triples, etc..). Same for other notions that I mentioned…

But then, those possibilities are possibilities so to say “outside” of the subject. All those possibilities involve possibilities of things which are not the subject himself (Neo).

The idea I want to propose now is that maybe Neo became aware of his abilities to become aware of those possibilities. The abilities to become aware of those possibilities are belonging to the subject himself, to Neo, which would solve the last issue.

However, I’m not sure if there is no some logical problem there. But this is best I can do, I think.

14 thoughts on “Can Brain in A Vat learn Math?

  1. Even a brain in a vat is having empirical experiences. If math is either a matter of structure or symbolic manipulation it does not seem to be the sort of thing a virtual reality would interfere with.

  2. Presumably the brain doesn’t distinguish between what is ‘external’ and what is ‘internal’. This is a kind of holistic distinction that is made based on the full set of experiential properties — in other words, for the brain there is no difference between an illusory rabbit and a real rabbit..there is only the full experience of the real rabbit versus, say the rabbit that is also pink and five feet tall that is experienced as a dream. This is part of the reason why one of the critiques of the BinaV is the sheer complication of actually tricking the brain…it’s so hard to trick the brain because the brain really isn’t inclined one way of the other…it’s the world that inclines one way or the other. This is also what makes possible large scale hallucinations. When you have something as real as the backdrop of the world included in your hallucination…we don’t have the characteristics distinguished well enough to tell the one from the other.

    Our knowledge of mathematical principles cannot be judged by the standard of our knowledge (or awareness) of objects. Mathematics is (arguably) linguistic in nature or are concepts or something along those lines. I can’t see any reason why the brain could not acquire the relevant concepts. They don’t seem to me to be contingent on the realness of objects.

    Perhaps there is a question here in which the case of mathematics is an example of, ..that is: how does the brain acquire concepts at all? Is some experience of possibility necessary? Is some experience of objects (real or not) necessary? It is at this point where the rationalist creeps in (having been waiting for the opportune moment all along like they always do!). After all — content without concepts are blind. Without some ‘internally’, i.e. innate, structure our experience would be just an ad hoc random non-understandable, even in principle, chaos. But, it’s not…and so the brain in a vat can acquire mathematics, as well as concepts of color and shape, because any sort of experience will do as long as there is the correct medium by which these experiences are brought together and ordered.

  3. Clark,

    I guess that is the issue for me – in what way are those experiences – empirical, in what way the subject learns something from them? In the general sense, yes, I think it is fair to say that BiV has experience, but how to make sense of the part that it is supposed to be empirical? The BiV isn’t causally related in a relevant way or in any representational way related to anything in the world, so that if we read “experience” in the externalist way, there is apparently nothing in that experience which is related to those things that, we seem to agree, BiV learns – like different notions of math, different general notions like thing, multitude, property, change, “other minds”, society, and so on…

    So, even if math is matter of structure/symbolic manipulation, in what way does BiV becomes aware of those things?

  4. Trey,

    Yeah, innateness of some of the concepts, or at least innateness of the form/structure through which we experience and which would give rise to those concepts seems as a very attractive position. But, there is that other problem – aren’t we getting into psychologism about those concepts in that way? Of course, some might be OK with that, and relate it to evolution or something, but to me it just doesn’t sound right to say that mathematical truths have psychological basis. (That’s what I meant by saying that some of us are not Kantians)

  5. The BiV isn’t causally related in a relevant way or in any representational way related to anything in the world, so that if we read “experience” in the externalist way, there is apparently nothing in that experience which is related to those things that, we seem to agree, BiV learns – like different notions of math, different general notions like thing, multitude, property, change, “other minds”, society, and so on…

    I’d disagree. The BiV is in a world. There will be patterns to its senses. If those patterns are mimicking real world phenomena it doesn’t mean that there isn’t a shared nature between the two. (Call that their iconicity) If math is primarily about patterns in possibilities then I just don’t see what the problem is. Mathematically if I’m counting and abstracting from the things I count to the notion of possible ways of counting then what is counted doesn’t matter. That is whether I’m counting virtual apples or real apples it is the same. That’s because math refers to pure possibility rather than actual entities. So the types of questions that get brought out in the externalist/internalist debate aren’t relevant.

    I should add that one of the early philosophers embracing a very empirical way of learning math was Peirce yet he was (contra most empiricists) quite the externalist. So all of this was actually worked out around than 150 years ago.

    One can dispute whether the empirical treatment of mathematics is correct. I tend to think it is. But I can understand those who say it misses the point.

  6. Clark,

    Of course BiV is in the world, and yes, the pattern of affection of the BiV’s senses is the same as when it is experiencing real-world phenomena(BTW, if we are talking about my scenario, I wouldn’t say that there is any mimicking, as it is merely a result of mere chance).

    I guess you are not saying that those notions are present *in* the pattern of affection of BiV’s senses. I can’t see how math notions, or others which I mentioned like society, communication, other minds and so on, can be present in a pattern of activation of BiV’s senses.

    Instead I guess the pattern is supposed to play some role of “pointing” to those notions. Is that what you mean by iconicity? But why would any pattern of activation of senses point to those notions?

  7. Yeah, mimick isn’t the best word. Peirce’s term iconicity is better.

    I think that from a Peircean perspective the iconic nature of those things entails they needn’t be tied purely to intentional uses. Thus there isn’t a problem. If we take math as normative possibilities then I don’t see what the problem is. As I understand what you’re getting at math is only math as intended by other individuals. That is math is fundamentally a kind of language game and that takes two people which is impossible given the way you’ve constructed your vat. (i.e. this is a vat without a conscious demon behind it)

    The problem I have with this is that while I think math is a kind of game I don’t think it is a game that can’t be private. That is I reject Wittgenstein’s private language thesis here. To me it’s all about signs and the effects they have. The trick with mathematics qua mathematics is that math inherently is as broadly general as possible. That is math as math doesn’t refer to anything specific. It is about possibilities as broadly construed as possible.

    Given that math as math divorces itself from particulars the nature of the vat is irrelevant since that is tying it to particulars.

    Once again whether one buys this depends upon how one views mathematical foundations. But there certainly are foundations for which one could learn math empirically but would have no problem with the vat.

  8. Just to add – consider intuitionism either of Brouwer’s sort of Dummett’s sort. These folks are anti-realist about math (that is it is tied to a human mind) but one could allow it to be learned empirically. Yes this empiricism isn’t of the outside world but for the intuitionist it doesn’t matter since all that matters is constructing the proof in the mathematician’s mind.

  9. Thanks,

    I agree that the BiV will “learn” math in this scenario, and as I pointed in the post, it does seem to me also that the way to approach this issue is to think about those notions (mathematical, and other abstract notions) in term of general possibilities and divorced from any real particulars, so I agree with you on that too. That would also fit nicely with the fact that we can figure out those things on our own (e.g. people figuring out math truths that they haven’t learn from anybody).

    However, where would the notions required for any such awareness come from in case of BiV? I agree that math truths can’t be in particulars, but then less so they can be found in the pattern of activation of the senses. You mention that they might come from the mind (if I understand you right) – does that mean that we should buy the idea of innateness of those notions?

    Sorry if I’m missing your point, I’m ignorant of the intuitionist approaches to foundations of math.

  10. I still don’t know what notions are missing in the vat. Could you be clearer here and explain why they are missing? I’m missing something. If there are patterns then there is mathematics.

  11. An other way to think about it is to say that for many figures mathematics is already a purely virtual pursuit so the fact it is being conducted in an empiricism also purely virtual should not detract in the least from what one does or what one means.

    I think you might find intuitionism rather appealing as it seems related to the project you are doing. Dummett is, of late, the most famous proponent although I find Dummett often muddled in how he casts his categories. (Including the very way he characterizes anti-realism) Brouwer is better but has a bigger flaw in that mathematics for him begins upon an intuition. (Thus the name) But the nature of the intuition is perhaps too mysterious and has oft been criticized.

    Of course I’d favor Peirce who offers some strong paralllels to Brouwer but with some key differences. Sadly most of the papers on Peirce’s conception of mathematics weren’t published until long after his death and weren’t really grappled with until after Brouwer’s position was already established. If you’re interested and have access to JSTOR check out Patin’s “Pragmatism, Intuitionism, and Formalism.” He does a pretty good job starting from the pragmatic maxim which binds all the varying kinds of pragmatisms together and applying it to math. (i.e. the meaning of something is wrapped up in the consequences of an operation)

  12. Oh, I’m not saying that those notions (numbers, colors, notions like – society, communication, other-minds, smell, taste, and so on…) would be missing, I actually CAN’T see any reason to think they will be missing (even given my externalist tendencies). So, I actually think that the BiV in the scenario will have those notions.

    So, we are agreeing that BiV will have those notions. BiV doesn’t have a problem, it is me who is having problem understanding where these notions “came from” :). Related to this of course the general question appears of what does this “having a notion” consist in, and probably a lots of other questions.

    Thanks for pointing that intuitionism might be interesting to me, I will do some reading on it. As for the paper you pointed to, I don’t have access to JSTOR, but probably will manage to get it.

  13. A useful category if one is an externalist is what Putnam and Chomsky have done. Putnam talks about narrow states where narrow mental states are states that depend purely upon the individual. Chomsky uses the same notion in many of his papers but simply adds a subscripted 1 or 2 to them depending upon whether they are narrow or broad and then says he’s just talking about narrow mental states. That way he can acknowledge the issue of externalism without having to grapple with it.

    I think this right. So, for instance, we can talk about the experience of a particular color. But clearly there is a narrow and broad (if one is an externalist) component to the experience. Narrow mental states will be identical whether one is in a virtual world or the real world. Broad mental states will be different.

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