HOW A PRIORI KNOWLEDGE IS POSSIBLE
Note 1: With Gauss I am one of those who is not at all impressed by Kant - and indeed Russell's possible reason for his greatness seems odd: he continued teaching philosophy while elsewhere there was war! How astounding! How philosophical! The question that heads the present chapter - "How A Priori Knowledge Is Possible" - indeed is a question Kant posed, and is itself an example of why I dislike Kant: I simply would have asked "Why are there innate principles of knowledge?", since I cannot make real sense of Kant's terms. And perhaps I should add that Russell himself, at least in his "History of Western Philosophy", also does not seem to have a high opinion of this philosopher. Back.
Note 2: It seems more correct to replace "Before the time of Kant" by "Leibniz claimed", for that is historically more correct. Back.
Note 3: In point of historical fact, Hume derived his ideas from Newton (who inserted his Rules of Reasoning in the second edition of his Principia precisely to deal with the problems connected with inductive and causal reasoning), and philosophers like Leibniz (in his Nouveaux Essays) and Autrecourt (in the 13th Century) also seemed to have arrived independently at the conclusions about induction Hume is famous for. Back.
Note 4: Kant may have "perceived" this, but if so he perceived wrong: the propositions of arithmetic and geometry can and have been be deduced from axioms in a deductively valid manner, and are not synthetic in Kant's sense. Back.
Note 5: Again, I want to avoid Kantian locutions like "How is pure mathematics possible", for it seems both barbarous and unclear. I much prefer a question like: "What is the explanation of the apparent validity and universal applicability of mathematical theorems?". It is true that this is not an easy question to answer sensibly. Back.
Note 6: Neither of Russell's reasons carries much weight, for it is also impossible to prove the validity of all deductive principles by deduction, so the weakness of induction (if indeed Russell's argument about induction is true, which has been denied by people like Peirce and Reichenbach) is no good reason to prefer deduction over induction, while it may be very much doubted that ordinary people can unproblematically and correctly infer mathematical truths from a single instance, even if some mathematical geniuses might, perhaps, be able to do so, in some circumstances.
The reason the empiricist's ideas about the foundations of mathematics do not make much sense is that they cannot account on their own principles for the kind of validity mathematical truths seem to have: on the empirical program, mathematical truths are merely empirical truths that happen to hold in very many different circumstances. This is something very few people who know anything about mathematics can believe. Back.
Note 7: With this, at least, I can agree, having argued the same in earlier notes. Back.
Note 8: This seems fair to hold about colours and tastes, but not at all obvious for space, time or mathematics, to say the least. And in any case there is the problem that what is contributed "by our own nature" may start to differ if "our own nature" starts to differ, either because we grow (or deteriorate) into something slightly different or because we come into circumstances in which our innate facilities start to work differently (which may happen because of drugs, illness, climate and so on). Back.
Note 9: In notes to earlier chapters I have commented on a similar mistake. The fact that all we have are experiences is not a reason to conclude validly that we cannot know anything but experience, for the same reason as we cannot infer validly from the fact that we express truths in language that the truths we express must be linguistic. (In short: our experience and our language are both symbolic or representative, and may stand for something entirely different from experience and language.) Back.
Note 10: These arguments are very similar to mine in notes 6 and 8. Back.
Note 11: This notion that what is "a priori" is "mental" is in large part due to Kant, and is one of my reasons for not liking Kant, for it seems an obvious confusion: even if human beings must think and feel in certain ways because of their constitution, there is no reason to confuse this supposed fact with a quite different possible fact, namely that reality has certain necessary features. Back.
Note 12: I am in fundamental agreement with Russell's line of reasoning, but it should be remarked that our belief in the law of contradiction and in anything else we express linguistically in fact involves assumptions about how language and what language may be about are related. Back.
Note 13: As pointed out in the previous note, matters are more complicated than this, and it is more correct to say that the law of contradiction is about the relation between language and what language may be about. Also, it need not be "a fact concerning the things in the world", except to the extent that given the conventions we use for the words "and" and "not" there can be no facts in any world we attempt to describe with these words and these conventions that correspond to a contradiction. Back.
Note 14: Indeed. It is a curious fact that there are to my knowledge no philosophers who have seriously discussed what may be called "the a priori falsehoods the human mind believes in", such as that "my leader is a wise and good man", "the group I belong to is superior to any other group", "conformism is virtuous" and many more in the same vein. Back.
Note 15: This is a strange notion and needs some explanation. The explanation is this: According to the mediaeval logicians, words can be divided into what they called "categorematic" and "syncategorematic terms", by which they meant that the former - like "elephant" and "green" - mean something by themselves, and the latter, like "and" and "in" mean something only when used together with categorematic terms.
In effect, Russell seems to be saying here that (1) all such human knowledge as is necessary concerns syncategorematic terms, (2) which do not have any meaning by themselves.
Given this explanation, Russell's notion may appear less strange, but it still seems to me to be false, since, for example, mathematical knowledge does not concern only syncategorematic terms (but properties and relations of things referred to by expressions like "straight line", "angle", "number", "circle", and so on). Back.
Note 16: Then in what sense does "in" exist in which I or my room do not exist? The only clear answer I can give is the one given in the previous note: "in" is a syncategorematic term, i.e. it has no (definite) meaning standing by itself. But this not what Russell needs for his conclusions, if only because it seems to be mostly a linguistic accident that a term is syncategorematic, and normally can be translated into categorematic terms, as when we say "Russell's room contains Russell". (Perhaps it is of some interest that the present passages used to puzzle me a lot when I first read the "Problems of Philosophy", when I was much younger and did not know about the distinctions explained in this and the previous note.) Back.
Note 17: It may be true that the many philosophers who held this followed Kant, but Kant followed Leibniz, who originated the doctrine (if Socrates or Plato didn't, much earlier). Back.
Note 18: Here Russell's strange notion commented upon in notes 15 and 16 gets used, and it is a mistake, that very strongly goes against a principle Russell espoused in many other contexts, namely the principle of parsimony, which says that one should not introduce more assumptions (to explain something one wants to explain) if one can do with less (assumptions to explain what one wants to explain).
As stated, it seems a pretty wild metaphysical assumption to introduce yet a third world, 'a world which is neither mental nor physical', and I much like to avoid it, not only because of the principle of parsimony, but because it is difficult to make sense of.
However, it seems likely that what Russell really might have meant is something like the following:
As I explained e.g. in Note 1 to chapter 5, our experiences are always signs or symbols of things that may exist or not. Now, apart from the fact that a given experience does represent a (non-)existing thing, it represents i.e. has meaning, which indeed we must understand before we are capable to judge reasonably on the question whether what is meant does or does not exist.
These meanings of our experiences - if we disregard self-referential statements and the like, for the moment - are generally taken to differ from our experiences, even if believe or know that the meaning of our experience does not exist. For in general experiences are experiences of something, signs are signs of something, and symbols are symbols of something, and in each of these cases the 'something' is meant, represented or what we think of by means of these experiences, signs or symbols.
So there is, in a way, a realm between experience and reality, namely the supposed meanings of our experiences (which, to remind the reader, may be contradictory or fantastical statements or pictures), which arises because every experience is taken to be an experience of something, and we are quite capable of saying or imagining what things should be like or should not be like so that our experiences could be true.
But it seems to me this realm of meanings, that may be alternatively described as whatever people may think of, whether real or fantastical, or possible or impossible, very much depends on the human mind, and consists of whatever hypothetical or conceivable things that are represented by our experiences, and which are, apart from their possible real existence, and apart from the experiences, signs or symbols we use to represent them, items in our memories, which indeed may well be called thoughts or ideas.
It is this realm Plato meant by his 'ideas' and hypostasized into Ultimate Reality. The notion of ideas, in the present sense of the supposed meanings of our terms and other experiences, is a very useful one, but there is no reason to believe it exists other than by supposition and elsewhere than in a mind.(It is quite simple, given familiarity with English, to think of such things as 'the impossible objects such as are dreamt of by gryphons and mermaids', and quite difficult to believe that the meaning of this phrase, which is easy to understand - indeed, not much more involved than 'the possible objects such as are dreamt of by politicians and crooks', the only important difference being that we can think of examples of the latter but not of the former - exists in any other way than by supposition, depending on the idea of representation and the known meanings of the terms in the phrase.)
Hence there is a third realm of things that are not (primarily) given experiences or real things, namely the things our experiences are experiences of: our thoughts or ideas - the things we think that are represented by the signs and symbols that make up our actual experiences. These ideas or thoughts may be false in that they are of impossible or non-existent objects, but they exist as the conceivable things our actual experiences and actual terms stand for, and they are made up of memories, including fantastical combinations of parts of memories (which may produce such objects of thought as gryphons and mermaids and their dreams of impossible objects).
My main difference with Russell is that for me this realm of meanings, of things represented by our experiences, signs and symbols, is mental and consists of ideas, which are creatures we make ourselves from selected parts of our sensations and memories by fantastical combinations of these parts, usually with the help of some schematism derived from language or mathematics (such as the phrases "the things which", "all things such that", and the simple juxtaposition of terms in predicate-subject combinations).
As far as I can see Russell denied that meanings are mental because he confused experiences and what these experiences are about, whatever it is, real or not. But the difference involved is the same as that between any given experience of a memory, that is given in a here and a now, and what it is a memory of, which must be something else than what it is given here and now, for else it would not be a memory, and indeed for me ideas are memories, whether of our sensations or of the fantasies we earlier made up by combining parts from sensations and parts from memories.
And the argument to assume that there are ideas is of the same kind as the argument to assume there is a reality: we may assume what we please to explain what is given to us in experience, and we assume that the reason that we see things appear and disappear again and again in our experience is that what we experience are sensations that reached us from real things - such as other people or our bodies - that existed meanwhile all the time in an independently existing reality, whether we experience them or not. Likewise, we assume that the reason that we can think and stop thinking again and again about many things, fantastical, real, dubious, possible or necessary as may be, is that what we experience are recallings that reached us from real ideas or memories - for example about other people or (what we think of as) ourselves - that existed meanwhile all the time in our minds, and which we were either born with or made ourselves earlier.
Finally, at this point the simplest hypothesis about minds, that avoids duplicating realities unnecessarily, and thus avoids one of the gratuitous assumptions both Plato and Russell made, is that minds are brains and that brains are part of the one reality in which we all live and try to survive; that ideas and memories are structures or processes in our brains; and that our conscious experience is an ongoing report of some of the states of our nervous systems including such states - of eyes, ears etc. - that inform us about the environment, and that enables the making of decisions about what to believe and what to do, so as to manoeuvre our body safely through a dangerous environment, where such decisions are made at least in part by joining present experiences to memorised past experiences and ideas about them, in order to make inferences about what may and may not be the case, which are stored in the form of altered ideas and give rise to our decisions about what to do and not do.
And the reader should realise that the above choice may be made differently, in favour of Platonic Idealism or Cartesian Dualism. However, apart from this choice, on any of these metaphysical hypotheses there are experiences of what is here and now going on in our mind and environment; memories including ideas of what went on and what we inferred about that; and something real - however fleshed out metaphysically - in virtue of which our memories and ideas are true or false.
Much more might be said at this point, but since this remark is already long I leave it at this, except that I recommend the reader who is interested in the philosophy of these problems to consult C.D. Broad's classic "The Mind and Its Place in Nature" where the present issues are discussed with great subtlety and clarity, and at length. Back.