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.

This is not a Meme!

After reading few posts on my favorite blogs lately, I though I just put a list of links to them as a sign of appreciation, respect (which turns into admiration in case of certain posts I’ve read on those blogs) and as a “thank you”.

So, in no particular order: Siris, Splintered Mind, Mixing Memory, Mormon Metaphysics, DuckRabbit, Philosophy Sucks!, Soh DAN, The Ends of Thought… I better stop it, before I name so much of them that the word “favorite” looses any significance.

Examples of Math being Empirically Justified

Richard at Philosophy Sucks! gave this example of how math truths might be empirically justified:

Suppose that you had two pens of sheep; one with 6 and one with 7 sheep. Now suppose that you counted the sheep individually in each pen (and got 6 and 7) and then you counted all of the sheep and got 14. Suppose you did it again. 1. 2. 3. 4. 5. 6. Yep six sheep in that pen. 1. 2. 3. 4. 5. 6. 7. Yep seven sheep in that pen. Then all the sheep. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Suppose that this was repeated by all of your friends with the same results. Suppose that it was on the news and tested scientifically and confirmed. Suppose that this phenomenon was wide spread, observable, and repeatable. If this were the case we would be forced to admit that 7+6=14 is true therefore mathematics is empirically justified.

I got to say I’m little disappointed with this example, because it doesn’t work at all… In the comment there I said that to make things fair for the rationalist, we should after counting the sheep from first pen, do both those things in same time: a)continue counting the sheep in both pens and b)count the sheep from the second pen.

So, we count the sheep from the first pen, 1. 2. 3. 4. 5. 6, and then for each sheep in the second pen we continue both countings – – so (7. 1.) , (8. 2.) , (9. 3.), (10. 4.), (11. 5.), (12. 6.), (13. 7).

What we did is then, that we counted all the sheep (13), and counted the number of sheep in both pen (6 and 7).

But anyway, I still have trouble figuring out what would it mean for math claims to be empirically justified. It is not like as if we can find mathematical entities in the world, so that we can test them. We could do this kind of counting and be surprised that everytime when we get 6 and 7 we get 13, but surely it is weird thing to do – given the we agree of how we count, it can’t be otherwise!

Another Vague Argument For Principle Of Sufficient Reason

P1. Only what can be fully understood can be partially understood
P2. If there is no sufficient reason for something about the world (any event, fact, and so on… in the world), then that something can’t be fully understood. (because understanding it requires understanding why it happened, why it is as it is, and the answers of those things are sufficient reasons)
P3. If something about the world is not fully understandable, then the world can’t be fully understood
P4. From P2 and P3 => If there is no sufficient reason for something about the world, then the world can’t be fully understood.
P5. From P4 and P1 => If there is no sufficient reason for something about the world, the world can’t be partially understood.
P6. We are bound to say that we partially understand the world.
P7. From P5 and P6 => There is a sufficient reason for everything about the world.

Two New Dogmas

1. There is such thing as phenomenal experience, which represents things in the world, which has what-it-is-likeness. There are also facts about it (phacts), and further it can be veridical or not (falsidical) which depends if the state of affairs it represents obtains or not.


2. There are such things as theories in our minds, and their base of our conceptual thought. There are facts (teofacts?) about those theories, which are there, and are independent from the issue if the theory is right or wrong. So those assumed things called “theories” are (similarly to the “phenomenal experience”) – false or true, depending on this – if the state of affairs that the theory represents obtains or not.

Oops, I just wanted to save this as a short concept, so I can work on it some later time. But I mistakingly published it. I won’t fight my subconscious,so I’m letting those two paragraphs with all their vagueness to hang here on the blog.

Why am I a Rationalist, and What I Mean by Rationalism…

Richard has argued against rationalism in few posts on his blog. I want to put forward few notes about how I see rationalism…

First, I think that rationalism is about understanding. It is a belief that some universal truths (set vs. the truths about individual events and things) can be in principle understood – that is, rationalism is the view that one can not just know the universal truths which obtain in the world, but also that one can understand the reasons why those universal truths obtain. I say in principle, because it easily might be that our reason is limited in that way, that we can’t understand everything (which is what I believe anyway).

It is a belief of rationalist then that there are explanatory reasons for some universal truths (I say “some” as rationalist can be rationalist about some specific thing, without being rationalist about every universal truth), and that we can become aware of that explanation. It is this which I think is important difference with empiricist – the empiricist as far as I can see denies one of the following a)that there are universal truths or b)that there are explanatory reasons for those universal truths or c)that we can become aware of those explanatory reasons. It is because of this, I think of empiricism basically as more pragmatic, or ultimately pessimistic stance, and I don’t believe there will be definite arguments of why one should prefer rationalist or empiricist stance. It might go even into personal character traits!

However I personally think that every philosopher has a little rationalist inside, because as far as I can tell they are all interested in understanding. And not just philosophers, scientists too. They are trying to understand the world – to figure out the reasons for these or those phenomena being as they are, or behaving as they do. (They are not very happy with “shut up and calculate”) And without assuming both a), b) and c) from previous paragraph, I can’t see how one can claim that there is a possibility to be understanding the world, or understanding some specific phenomenon.

But to some other thoughts…

First, I think that rationalism is compatible with empirical research. That is – there is nothing wrong with path to understanding being through empirical research. I think even that given the fallibility and general limits of our reason, that empirical research and empirical confirmation are to be preferred and required by rationalists too. The requirement for empirical confirmation, being able to give predictions which can be confirmed empirically, doesn’t mean that the prediction have to be based on empirical research. The common sense requirement for good predictions, and need of empirical confirmation, is separate issue from how we did cam to those predictions. Given the sense of rationalism that I pointed to, rationalist expects that it is possible to understand the phenomenon, and based on this understanding to give good predictions (even they can be fallible). One can on this issue, easily compare a method which may be used by mathematicians to discover if the theorem is good or not. One can just take few specific cases/numbers, and check if the theorem is really OK. If there are some specific numbers for which what the theorem states doesn’t hold, there is something wrong with the theorem. And with the thinking behind its proof. Of course, this is not perfect analogy, but it might show how I think that different venues of research are compatible with the idea that we can understand things.

Other thing is relating rationalism with experience in general. Given that understanding will probably require awareness of things, it is normal for rationalist to accept that experience, or becoming aware to be necessary requirement for the ability for a priori thought. Even in cases like Kantian rationalism, even the forms of perception and pure concepts of understanding are innate, we don’t become aware of them in any other way but through experience. This will be even more so for other kind of rationalists (like me) which think that we become aware of things in the world through experience, in which there appears identity between our idea of those things and the things themselves. So to say, what we become aware, and what we have in mind are the things themselves (the notions). It is clear that this presupposes the great role of experience, as something through which we become aware of things – about which later we can think.

Third thing about rationalism is that when put vs. the holistic models like Quinean one it seems very simplified. But, it should be clear, from how I described rationalism, that there is nothing which goes against holistic view on knowledge, and really also on metaphysics. Even, seems most natural given this view on rationalism, to say that the ideal goal of understanding is to understand everything, and that full understanding can be reached only by understanding everything. And that what we in fact gain is just partial understanding. However for a rationalist like me, even partial understanding is possible only if the world is in principle understandable (of course, it doesn’t follow that we do in fact have abilities to do so. Personally, I doubt that).

Fourth thing which is usually brought against rationalism is the non-euclidean nature of space time. It is pointed that based on a priori thinking Kant has came to wrong conclusion – to the assumptions that the phenomena in space time are best described through euclidean model. But, one can point that a)in the years that followed Kant, Hegel has argued against this taking of space and time as separate entities, or as forms, and b)that the Einsteinian conclusions regarding space and time are in big part based on a priori thinking. (thought experiments, math, logic, and so on). And, if that is not enough, one can point to the non-euclidean geometries being developed within the math, which is a discipline where we know how things are settled – not through research of the world, but through rational arguments. How else could the abstract questions of math be settled? We can’t really find numbers or abstract spaces in the given abstract form in the reality. To those that deny the importance of this form of thought, one should just point to the number of theorems that math has already came to, which didn’t have any use at the time, only to find use later in relation to some less-abstract phenomenon.

And fifth, rationalism doesn’t presuppose existence of some magical entities like numbers or “pure forms of understanding” or Platonic ideas. For me it means understanding of the world, and the phenomena of the world as far as they might be taken to fall under certain general notions. To give an example, I think that 1+1=2, is properly understood as “whenever there is a pair of things, there is one and one more thing”, or “whenever three objects form a right angle triangle, there will be such and such relations between those things”, and so on… Are those truths about the world (given the idea that whatever is necessary truth it is not truth about the world)? Well, yes, as far the world falls under those abstract notions. Given the holistic take that I mentioned before, a rationalist can of course deny that those abstractions will be perfectly applicable to the worlds, and argue that there will be something always missed. But *as far* things fall under those notions, I don’t think there is a reason to deny that those truths are about the world.

Figuring Out Things

There are two methods of figuring out things about the world. One is by checking the world, and the second is by thinking given what we already know about the world. Say, that we want to figure out if there is a beer in the fridge. If we know that there weren’t any beers in the fridge yesterday but that we went and bought a pack of beers, and if we know that we already drank 11 beers, we can do the calculation and through thinking figure out that there is one beer left in the fridge. Or, alternatively, we may get up and check the frakking fridge.

In general, in everyday speech, we use the word ‘logic’ to characterize the cases where we figure out things by thought -How did you know that? -Logic! . More specifically we may use ‘math’ to speak about certain cases. Like in case with the beers -How did you know that? -Math!, though even in that case it seems more natural to just say -Logic!

Outside of everyday speech, ‘Math’ and ‘Logic’ are more used to refer to social disciplines, which are worried about abstract form of this kind of reasoning, or to refer to those abstract forms to which those disciplines had figured out. Leaving out the specifics, they try to give ready formulas which could be applied in any case where the abstract notions which are part of those formulas can be applied. So, 12-1=11 in math says that whenever there is 12 things, they can be divided to 1 thing and 11 more things. Be them beers, roses or toasters.

Besides those disciplines, we have scientific disciplines, which are about two things – a)explanation and b)prediction.

As far as sciences (or you) are interested in explanation, the prediction is not enough. Just that one can predict something, doesn’t mean that one has understood the thing. I may predict something, by reading out the numbers written on computer screen, because some person who understood the phenomenon has written a program which can predict how the phenomenon will behave. But because I can predict what will happen doesn’t at all mean that I understand the phenomenon. However the prediction has more significant relation to understanding in another way. As we can make mistakes in our thinking, we may be not sure that we understand the phenomenon. In such case being able to give good prediction is crucial to us figuring out that we didn’t make some mistake in our thinking.

This part -being able to predict is crucial part of scientific method. But does this mean that the sciences are something like checking the fridge to figure out if there is a beer there?

For sure not:

1.Predictions are result of having a theory in a first case. In one case this involves figuring out a model, from which the previous data we took from peeking into the world will follow. So, one has to have the logical thinking (in the general sense) in the background, for the whole notion of having model from which empirical data will follow to make sense. In another case, one will start from the empirical data, and logically get to the theory (like in the case of Einstein’s Relativity).

2.If the predictions are supposed to be novel and good, they should be as far as possible from the previous empirical data. Where do they come from? From logical thinking – they are supposed to follow from the theory itself. One is supposed to get to them by thinking.

3.As far as we want to understand the world, we won’t be satisfied only with a statistical correspondence. In that sense, one who cares about understanding, will not be satisfied with the predictions alone. And why will one call himself a philosopher or a scientist for that matter, if one doesn’t want to understand the world?

Besides this, checking the fridge is not something which is incompatible with things that we should know based on thought alone. In programming, in lot of the cases, could know what the program will do in different cases. And when we write a program, we write it in such way that it would have that certain function that we want it to have. So if we are to predict how it would work, we would predict that it should work as it should. But, given that our thought is fallible we get such things as bugs, and we do tests and debugging even the functioning of the program should be in principle be knowable from mere knowledge of the code. And mathematicians can test the truth of their theorem by trying out if it really works on some cases. But what we get in this way, is not as much belief that the theorem is true, but that we didn’t make mistakes in figuring out the truth (or in logic of the program).

Separate from the discussion of prediction, one could also point that the abstract mathematical forms were already in place when science figured out that it can use them. So, it is not like those mathematical forms were always created in order to give a theory, that is from pragmatic reasons. There was no need for those theories when they were figured out. So, even if one wants to talk about value of certain way of thinking, it is clear that a priori thinking which wasn’t based on empirical checks, nor on pragmatic needs, showed itself as having an enormous value. In any case, it seems clear to me that logical (or a priori) thought is very important part of empirical sciences. They go beyond peeking in the fridge.

But, anyway, if one talks about thinking vs. peeking (rationalism about X vs. empiricism about X), it surely depends on the person and on the issue. If one is interested in understanding or knowing. If one is interested in understanding, one needs to figure out answers to why? questions, and why questions are answered by figuring out reasons, and reasons are what is figured out by thought. Peeking helps, but giving a list of states that follows each other isn’t understanding, without understanding why they follow each other.

A Link To Think

Peter Davis has a several funny cartoon videos on the history of philosophy, called ‘three minute philosophy‘.

Here is one of them…

Also… Don’t forget to check out the new philosophers’ carnival over at Big Ideas.

BTW, you probably noticed there weren’t much posts here lately. I suspect I did something bad to the muses, or to some close relative of their, so don’t expect too much from me in the following period.