.....SIX KINDS OF PROPOSITION - Sections 6-8
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6. MATERIAL AND MENTAL PROPOSITIONS
Language comprises, pre-eminently and primordially, a hard core of adventitious, material (or physical) propositions, the bricks and mortar of everyday conversation and of all science. Even though most material propositions are instantly recognisable as such and uncontroversial, the boundaries of this class remain vague, no satisfactory account of the criteria by which their truth is to be evaluated having yet been devised. Nor will any be offered here. My chief purpose is to sustain the "common sense" view of existence as promoted, for example, by G.E. Moore (1953), and doubtless embraced tacitly by the vast majority of ordinary people. According to this view, the world consists primarily of two distinct types of entities - material objects and "ideas" (or "acts of consciousness"). One might therefore expect there to be two types of proposition, material and mental, dealing with entities in each category. Support for this outlook comes both from its practical indispensability and from the fact that historically much of the labour of philosophy has been aimed at bridging the gap between these two realms, so far with little success. Far be it from me to revisit this ground by pulling out the major highly academic (and often needlessly cumbersome, if not plain needless) arguments for and against various kinds of mind/brain dualism.
It is probably misleading to suggest that there exists a "common sense" notion of mental propositions, since very few people are disposed to think objectively about matters of the mind. Mental experiences come in a very wide range of colours, encompassing such varieties as bodily sensations, percepts, creative concepts (including uses of language), emotions, memories, hallucinations and so on, and the boundaries between them are exceedingly hazy. I, for one, find it difficult to make head or tail of this lot, so in what follows I shall consider mental propositions relating to only two kinds of experience - bodily sensations and percepts (predominantly sensory information originating outside the body).
On the other hand material propositions, though just as diverse, readily succumb to a "common sense" point of view. This rests on the belief in a world of physical phenomena which are either "directly" detectable or whose existence is inferable from observational evidence, and which exist independently of any person's awareness of them. The strength of this belief is evinced by an unquestioning reliance upon one's personal capacity to relate to and interact with this world and these phenomena. Thus an elementary material proposition, such as "There are cod in the river Murray", is simply one which, if true, is representative of some fact which is presumed to exist in the real world whether witnessed or not. (I won't indulge in speculation concerning the existence of objective facts about which no one ever could, even "in principle", acquire observational evidence.) In addition to elementary propositions, however, there are others, such as "If the rain stays away the Socceroos will win" and "The cat's in the cupboard and share prices are up in Tokyo", which appear to be composed of propositional elements joined by logical connectives. I shall propose in §7 that these are either wholly experiential or else not propositions at all.
Even the most rudimentary sorts of material proposition, however, seem to perform a variety of functions. For example, one might distinguish between the following kinds:
(1) incidental - "The kangaroos are eating the avocados"
(2) attributive - "This sheep is a purebred Merino" (given that "purebred Merino" here behaves descriptively and not as a term whose meaning is being delimited by exemplification)
(3) actualising - "Non-polluting automobiles exist"
However, the differences between these kinds are quite innocuous and they are readily interconvertible (considerations of time and generality often decide which form is the more appropriate). I draw attention to them chiefly because of the risk of confusion with non-material types of proposition (e.g. attributive with characterising) .
While some philosophers have sought to defend material propositions against the inroads of metaphysics, others have contrived to distinguish them from statements about purely mental phenomena and so, perhaps, to repudiate those doctrines that embrace some sort of reduction of physical to mental processes or vice-versa. It seems to me that the "pre-scientific" body of metaphysics which properly educated people recognise as consisting of nonsense and falsehoods has received due punishment, to the detriment of much needed honest endeavours in "post-scientific" speculative metaphysics. In what follows, therefore, I wish chiefly to add my support to the physical/mental distinction, though not by assent to the hypotheses apparently most favoured by academics.
6.1 Investigability (verification)
(I do not now think these remonstrations carry much conviction. See final 2 paragraphs.)
Although the layman may well feel no difficulty in distinguishing his own thoughts from the facts of the external world, a little reflection suggests it is a fuzzy line that runs between the collection of propositions variously described as objective, physical, material, external, independent of consciousness, public, investigable or fallible; and the collection described as subjective, mental, private, internal, immediate, basic, uninvestigable or incorrigible. Using familiar arguments somewhat sparsely and in piecemeal fashion, I shall attempt to dismiss many of the shades of distinction implied by this colourful assortment of adjectives, beginning with the doctrine of empirical verifiability or, as I prefer to call it, investigability. In using another name, it was my optimistic intention firstly to encompass classical verificationism, its pragmatist siblings and its offspring, and secondly to emphasise that the attempt to verify (or falsify) is suspect, as much as the possibility of success. I take it that underlying the various forms of empirical verificationism there's a vague principle which seems to rely on the assumption, roughly, that worldly facts can be observed only by worldly means, and if this principle founders then of course the entire brood goes down with it - notwithstanding that there may be other reasons for thinking any particular adaptation of the doctrine objectionable.
The usefulness of the investigability principle might well be questioned in view of the great diversity of propositions whose objectivity and truth assessability we are willing to respect. We need only reflect upon the pragmatists' notions that we buy truth "on credit"; that much of our objective knowledge is second hand; that propositions are often the only evidence we can muster for the existence of the facts they depict; that most of them occur as unique utterances whose truth we take for granted; and that many quite ordinary looking propositions appear to be truth functionless and consequently, some would say, unverifiable. In this disarray, however, we shall find no good reason for rejecting the investigability criterion. Nor should we be unduly troubled by the threat to verificationism of the alleged theory-dependence of scientific facts (a grossly overworked tale), or by the charge that positivist verificationism suffers irremediably in the hands of positivist antimetaphysicalism, or by the more explicit tenet that no proposition can be confirmed by particular experiences owing to the universal nature of its descriptive terms (a suggestion that begs the question whether language can ever speak of particulars). Finally I shall ignore the application of investigability to universal hypotheses, as it seems to me that these are not, when taken literally, sensible propositions at all (see §9). As well, I have suggested elsewhere that there is no significant difference between a fact and a theory (see endnote to this "Footnote" on Creationism).
No, the reason why the investigability principle cannot advance our present purpose is just that it lacks any independent explanatory power. Because the world of objective facts is precisely the world in which people live, it's tritely true to say that a fact might have been verified had someone ("the resourceful observer") been in a position to investigate it. But to say that that is what constitutes its being an objective fact is circular, since the proposition that someone is in a position to investigate a fact of this sort expresses a fact of a similar sort, and must be similarly amenable to investigation. The possibility of verification, corroboration or falsification is contained within every presupposition of objectivity, and cannot prop it up from the outside. Verification moreover implies evidential checking of either a pre-existing factual statement or a pre-existing hypothesis or belief. But many propositions are individual reports which either defy the possibility of, or carry no expectation of, subsequent confirmation: the witnessing of the fact precedes the statement, the statement merely reports the fact as experienced and the experience does not verify the statement.
Furthermore, the supposition that objective statements must be capable of objective verification might well be considered pointless unless they were in need of verification. From this standpoint, an implication of the verification hypothesis is that such statements are wanting - that they somehow lack meaning and truth unless verified. For example, supposing you told me that yesterday you shook hands with the Prime Minister and I said I believed you. What, exactly, would I be believing? If the statement were in need of confirmation, not only would it be unbelievable: it would appear to be incomprehensible. But while it is undoubtedly a requirement of certain "difficult" propositions that they be evidentially checked to bolster their credibility, this is by no means true of the vast majority of factual propositions, which must and do stand on their own merit. Reliance on personal reports is an uneliminable feature of dialogue about the outside world. And although, at the sub-philosophical level, appeal to investigability is often necessary when the practical and social consequences of material statements are significant (I make ample use of the criterion in this article), as often as not the events recalled by such statements have archival or narrative value only.
Still, verificationism has done a great service to philosophy, and its enthusiasts probably discarded it too easily under criticism, not all of which was justified. Its claim to distinguish between meaningful and metaphysical statements was not pushed far enough and I believe it had more substance than is now usually accredited to it. For example (and in absurdly simple terms), electrons, radio waves and gravity are meaningful concepts, if not "real" entities, because they have highly predictable, useful effects in the real world - television, computers, nuclear power plants and space missions would be impossible without them. On the other hand God the holy ghost, demons and souls are metaphysical because they have no predicive power at all. But verificationist theory became too cluttered with ifs and buts. I now sometimes wonder whether all this complexity could be replaced by a simpler idea. If material propositions often seem to be endowed with some kind of necessity or certainty, doesn’t this ensue from the conviction that the facts related by historical propositions cannot be undone? Captain Cook has sailed and that's that. Electrons have worked miracles for medicine, who can deny that? Perhaps the verification hypothesis is essentially just a way of saying “Let’s wait and see what time tells”. Isn't that good enough?
Well maybe not. It’s useful to contrast verifiability with falsifiability. The reason why falsifiability is nonsense is that the propositions that are supposedly open to falsification are meaningless and the reason they are meaningless is that they are unverifiable! I’m referring, of course, to general propositions of the “all ravens are black” variety. (Also see §9.4 paras 2 & 3). If the possibility of verification cannot even be envisaged, the proposition is empirically meaningless. Verifiability holds an elevated position over falsifiability because meaning, truth and the possibility of being shown to be true are all closely allied, intrinsic elements of propositions.
6.2 Private and public
Even though the investigability approach be rejected, one might still think that objective or "external" propositions must be publicly accountable, in a way which apparently distinguishes them from subjective or "internal" propositions (such as "I have a toothache", as well as "You have a toothache"). Every internal proposition, it might seem, can properly be confirmed or denied only by the particular individual to whom it relates. Other people must judge the proposition on the testimony of the subject undergoing the experience, or else by physiological or behavioural investigations which might allude to, but seemingly cannot ascertain, its truth (see 'Mental and physical' below).
That a person can have a toothache and refer to just that experience by the statement "I have a toothache" cannot be seriously questioned. Nor would it materially alter the nature of the case if it turned out that some people have some internal experiences in common, whether through extrasensory perception or, conceivably, because their nervous systems have been connected in such a way that two or more brains register the same impulses. In a certain sense it seems possible to have someone else's toothache. But in addition to propositions referring to these bodily experiences, this class might be thought to include others, each of which refers to an evidently physical situation observed by a particular individual but which could not possibly be observed by any other individual. Thus the proposition "When I went for a walk by myself in a remote forest on this date last year I picked a wild apple from a tree and ate it", even if somehow publicly useful, is arguably not even in principle representative or misrepresentative of any publicly accessible fact; this statement, it might be thought, reports a private fact, although of course (no less than "I have a toothache") it is couched in the terms of a public language*. Besides these, however, a much larger number of factual reports might be said to be "contingently private", either because they just happen to represent unique observations (of publicly accessible facts) by isolated individuals or because the presence of other onlookers is inconsequential.
* Modern technology is getting closer and closer to rendering this kind of privacy obsolete. As we advance into the
This line of thought advances the view that the integrity of material statements is not compromised by lack of public access to the particular facts that they convey: there are statements which provide objective information; which are private in a similar sense to those which record immediate bodily sensations; and which earn instant recognition as describing facts in the physical category and not in the mental category. The objectivity of the material statement is thus independent of the number of people able to substantiate it - much less the number who happen to hold it true (for nonsense and falsehoods are quite as popular as truths). The number of person-observations required to establish a fact depends on the complexity of the fact, its "historical background" and relation to other facts, the difficulties encountered in assessing it and the kinds of project which it is expected to service. Over and above that minimum number, which in the majority of instances is just one, additional witnesses can serve only to vouch for the integrity of previous observers and not the nature of what is observed. Other people do not play an important role in determining the objectivity of our experiences: quite obviously, most of the time we are very capable of judging for ourselves and of acting accordingly. So, although it must be conceded that much of our objective knowledge is acquired through trust in reports, witnesses of reports and witnesses of witnesses, our basic concept of the material world includes the idea of a public domain and does not rely on it.
future the past becomes more sharply defined. In principle, nothing is hidden from public scrutiny. Nor is there any
good reason to suppose that our innermost thoughts too are inviolable.
The distinction between subjective and objective propositions is possible, one might surmise, because often there's a noticeable time-lag between the publicly accessible cause of an experience and the private effect - the having of the experience itself; thus it seems possible to refer to the psychological effect without implicating the cause, and vice-versa. And one might be inclined to assume that these two aspects always do retain distinct identities, even when there's no discernible interval separating them. On the other hand it's also possible to herd causes and effects (if that is what they are) into one fold, so to speak, though the name of the fold depends on the manner of herding. Suppose, for example, that whenever one saw a black cat one immediately felt a sharp pain in the head. Wouldn't one then call black cats "headachy" cats, thereby bringing the headache into the same objective camp as the cat? Maybe, or maybe not: depending, perhaps, on factors such as the ubiquity of the black cat/headache phenomenon and the degree of spatial associativity attaching to headaches under these hypothetical conditions. But whether or not there's a delay between physical cause and mental effect, it's possible to regard sensations and affections as being public and investigable in the following way:
(o) I enter a room full of smoke; next day I have a headache. Whoever enters this smoke-filled room will before long have a headache. (This is a headachy situation.)
(p) I look at the sun; I'm dazzled. Whoever looks at the sun is dazzled. (The sun is dazzling.)
(q) I look at a poinciana bloom; I have a sensation of orange. Whoever looks at the poinciana bloom will have a sensation of orange. (This poinciana bloom is orange.)
(r) I look over there; I experience an image of a tree-like form combined with sensations of green, grey and especially orange... Whoever looks over there will experience an image of a tree-like form combined with sensations of green, grey and especially orange... (The poinciana is in flower.)
Despite the traumas suffered by language here in performing the duties required of it, the ease with which one can draw up lists of graded analogies of this sort appears to provide considerable support for the view that if we are to make a clear distinction between subjective and objective propositions it will not be on the grounds that only the latter are public. And, as we have seen, nor will it be on the grounds that only the former are private. Both types appear to have both public and private aspects.
6.3 Immediacy and incorrigibility
Still, one might think, mental phenomena surely convey a sense of immediacy and incontrovertibility which is absent from physical events. This notion presumably supplies the incentive for the theory that material propositions can be reduced to, or constructed from, "basic" (or "protocol") statements describing personal experiences: that is, a statement such as "The poinciana is in flower" can somehow be translated into logically equivalent assemblages of statements about private sensations, which (it may also be alleged) are "incorrigible" in the sense that we might suppose "I'm in pain" to be; and therefore, possibly, that a statement which can be doubted is reducible to statements which cannot, at least by someone at some time.
This is a point of view to which the examples (o) to (r) might be thought to lend some credence. However, I presume the first (reductionist/phenomenalist) aspect of this position is no longer widely considered to be defensible, for well-known reasons. But even if reductionism were tenable, it could not achieve its apparent objective of transforming a statement which is allegedly on probation into ones which are allegedly unimpeachable. For if the original material statement is dubitable and its reduction indubitable, then its reduction is surely incomplete: while it may take account of sensory experience and perhaps spatial relations, it doesn't properly accommodate a variety of other relations and external factors. And if it did, we should soon see that the original statement and its translation are incorrigible (or not) to the same degree.
If feeling certain has anything to do with certainty, however, I'd be inclined to dismiss any further discussion of this point, since, while I write this essay, I cannot assent to feeling any less certain that there's a flowering poinciana outside my office window than that I have a headache. Nothing seems to be gained by rebutting with arguments about hallucinating or anything else designed to destroy my confidence about what I am perceiving. In this instance, anyway, no one could convince me that my experience is not of a genuine physical object. They might persuade me that I have incorrectly identified the tree or misnamed the genus, or that the poinciana is a fake - but not that it has no material existence. If I'm dreaming now, then life is nothing but a dream. Life depends on certainty at this level - and death frequently results from the imprudence that spurns it. But while I'm disposed not to question whether there exist certain experiential facts which I believe the statements "There's a flowering poinciana outside my office window" and "I have a headache" adequately describe, it appears to be quite another matter whether these descriptions too are reliable. In fact this plea for certainty has only a very peculiar point to make, namely that its claims should remain dependable even when inexpressible.
It is of course possible that another person, sitting next to me in my office and apparently looking in the same direction, should deny the existence of the poinciana. And although in that case I'd spare him not a little more scorn than if he had denied the existence of my headache, still, I would accord him the right to doubt whether I have a headache, and I ought to accord him a similar right to doubt that there's a poinciana over there, even when I feel certain there are no factors that might lead him to interpret the situation differently from the way I do. It is indeed central to my overall thesis, as I submitted in §3, that two apparently incompatible material statements (such as "There's a poinciana growing a few metres away from this window" and "There's no poinciana growing a few metres away from this window") are not logically contradictory but only experientially opposed: which is to say that although we may not now be able to imagine any circumstances in which both are true, the possibility of such circumstances arising remains open. But the assertion of this possibility carries no weight, it might be thought, unless it can also be claimed that both statements are infallible, as otherwise one could presumably find cause for discounting one of them.
This expectation of a cast-iron guarantee for the truth of the verbal concepts related by constitutive propositions is, however, just as gratuitous as the penchant for logical necessity which it replaces. It is thus not my view that there exist propositions which are incorrigible in some favoured, bullet-proof sense, but only that in the personal confidence stakes some of those we take to be objective are at least as safe a bet as the most intransigent of the subjective kind. Even without the buttress of incorrigibility, the consequences of this position are significant. For if some of the facts we normally call "physical" are just as indubitable as many of those we call "mental", then neither the conventional distinction between these categories nor the classical relations among physical phenomena can be upheld. We shall find ourselves unable to agree upon a method of deciding which of two or more incompatible sets of observations records the true state of affairs: which is only to say that no classical state of affairs is represented by all the observations jointly.
Karl Popper (1959) crisply dismisses the quest for verbal incorrigibility with the proposal that, because the terms employed in all statements are universal, our knowledge of particular facts can never justify the truth of any statement. At face value, this appears to imply more than perhaps it ought - that language by its very nature can never get to grips with particulars, that it can't really talk about particulars at all. Although this seems quite wrong to me, still I suspect the point Popper intends to make here is broadly correct - a point which crops up everywhere in different guises and which I touch upon in §9.5 and §9.6 in discussing attitudinal statements and fiction. In the present context perhaps it could be put like this: it is only in the act of contemplating our experiences that we face the problem of incorrigibility, a problem that seems to me to disappear as soon as it is confronted. For, the moment we start to think about our experiences, we create opportunities for doubting them. Even if we admit the possibility that thinking about experience doesn't entail using linguistic concepts, it surely involves a generalising faculty of some sort - the capacity to recognise particulars as being of certain kinds. And it seems to be of the essence of recognition that it leaves room for doubt. I think this is also the main point made (in a much more roundabout way than Popper) by Austin (1946). For my part, I'm fond of the two epigrams: No perception without conception and No pain without brain, but these imply nothing about certainty.
In so far as they capture some condition of the mind, all propositions might be thought to contain elements of "incorrigibility". I remain puzzled by claims that these elements are utterly indubitable. If we concur with the view that language contains expressions which are bound this tightly to experience, we shall surely be driven to the further conclusion that we cannot even have the experiences which the incorrigible propositions depict unless we do formulate those propositions; which is to say that whoever cannot speak, or, at least, think in broadly linguistic terms, has none of the common feelings and experiences which the rest of us say we have. For my own part (as a chiefly non-verbal dreamer) this conjecture seems preposterous: even if, as I think, perception is impossible without conception, experience is by no means impossible without language.
Many other objections have been raised against incorrigibility, both experiential and verbal. Some philosophers claim we can be mistaken about our own conscious states for reasons that include misunderstandings about how we really feel, inconsistencies between our apparent intentions and our behaviour, and plain self-deception. To my mind most of these arguments either miss the point or they are radically mistaken. In the end, it seems enough to remark, first, that most if not all of our sensory information is "interpreted": we become aware of it only after it has been processed by machinery and supplemented by data which were not acquired along with the (alleged) "bare" sense impressions. And secondly that no propositions can be formulated in the absence of such interpretation. It seems far from certain that even a mental grimace could be taken to be an "incorrigible" expression. Assuming it could, however, the next question would be whether "I have a crashing headache" says anything more than the grimace. If it does, I can doubt it. But it seems clear that every factual proposition, private or not, mental or physical, is in this position. The notion of incorrigibility cannot supply a criterion for differentiating between types of experiential proposition, since none possess that property.
6.4 Mental and physical
Then what grounds are there for the popular distinction between statements about our own experiences and feelings and statements about material objects and events? It might be helpful to return to a simplified version of the table used earlier on:
....................Objective language:.........Subjective language:
Mental.......A. This is a headachy..........B. I have a headache
Physical...C. The poinciana is...............D. I experience an image of
topic................in flower..................................grey, green, orange etc.
Although language can be modified according to whether we wish to emphasise the public consequences of a subjective experience or the personal aspects of an objective phenomenon, in practice it's difficult to imagine circumstances in which there's any more use in replacing B by A than there might be, for example, in going full circle with C and D by substituting "This is a flowering-poinciana-image-causing situation". The ungainliness of the latter expression doubly underscores the fact that excellent linguistic mechanisms have evolved to deal specifically with mental and physical topics, reflecting the natural division between these types - a division which to my mind (but apparently not to every one's) seems quite adequately exemplified by the glib remark that it makes sense to say "I was bashed on the head but was unaware of it" but never to say "I had a crashing headache but was unaware of it".
It is, of course, a stubborn peculiarity of mental phenomena that, in the conventional domain of which we speak, they cannot be properly described out of sight of physical phenomena. Every mental statement apparently refers to a mental event, implicitly or explicitly bound to an individual, a time and a situation. And although it's by no means clear that there are no mental phenomena that are not confined in this way, the language of western culture at the present time barely entertains this possibility. The role played by these physical accessories in simple mental propositions, however, appears to be purely presuppositional: for example, in "Lucy has a headache" the issue is whether or not Lucy suffers a headache, and not whether it's Lucy or Len or some one else that has the headache. On the other hand many conventional experiential propositions assert, rather than merely presuppose, physical facts in parallel with the mental. It can hardly be doubted, for example, that part of the propositional purportment of "Lucy had a headache while she was at the farm today" is the fact that Lucy was at the farm today. So the proposition could only be regarded as wholly true if (1) Lucy was at the farm today (which is apparently physical) and (2) Lucy had a headache (which is apparently mental) and (3) she had the headache while she was there (which alludes to the puzzling link between the two types).
One might suppose that the important question here is: how is it established that Lucy's headache occurred at the alleged time and place? - anticipating that the problem might succumb to either a wholly material or a wholly mental solution. If Lucy herself was aware that she was at the farm today whilst having a headache, and a personal statement from her is all there is to go on, one might get the impression that the latter is wholly mental, since it is only in virtue of her own psychological cohesion that the proposition has sense. This response, however, is at odds with my conviction, alluded to earlier, that the one-off reports of individuals about their experiences of reality are self-sufficiently physical. (Which is only to say: Lucy is just as capable of objective observation as anyone else!). That being the case, if Lucy utters the statement in question it is irrevocably mongrel in character.
If, on the other hand, time and place are determined empirically and independently of input from Lucy, and the presence of the headache is similarly established, for example by means of behavioural or physiological observations, then the statement that Lucy had a headache might seem to be entirely material, bearing no reference to any person's feelings. Here again, though, I very much doubt that the immediate method by which the presence of Lucy's headache is established determines the quality of the proposition(s) reporting that fact. For there appears to be no real difference in the quality of the verdict no matter how it has been reached. That is, there's no palpable difference between the statements "Lucy has a headache" (this having been established by scientific investigation) and "Lucy has a headache" (because Lucy said so). Surely the proposition (that Lucy has a headache) is mental even though it has been inferred entirely from the results of empirical investigation. Nor is the diagnosis any less reliable just because it isn't the direct report of the subject. Given an adequate scientific and communicative protocol, one might well find reason to place more trust in a judgment about the subject's state of mind inferred from the record of an independent scientific investigation than in the subject's own report. Thus the immediate methodology alone doesn't determine the quality of the proposition. But this conclusion is of little significance, as it's obvious that the immediate (empirical) methodology alone never does supply the entitlement to infer anything about anyone's state of mind. This entitlement derives, rather, from the combined historical power of an elaborate scientific wisdom, close communication with other people and personal experience of phenomena such as headaches. There's more to methodology than meets the eye. (Also see §9.5 on attitudinal statements.)
I might have begun this discussion with the naive proposal that physical objects and events are extended in space while thoughts are not. Today, I presume, almost no one would disagree that this embodies a serious misunderstanding of the relationship of these categories. The amusing regalia of modern technology have conspired to ensure that there are few who cannot interiorise spatial ideas to whatever degree suits their purpose. It is not just that we can speak as if external phenomena are internal, but that we can, at least when in a state of passive observation or aesthetic reverie, mentally align physical objects and events with our most immediate sensations and affections. Surely it's enough to remind ourselves how easily we can be tricked into believing that certain phenomena have concrete existence, in a sense in which they really do not. But while the senses of sight, hearing, touch, balance and so on supply us with particular impressions of space, which seemingly bear no relation to one another except that they are taken to represent aspects "of the same space", and can individually mislead us with regard to the presence of objects or the occurrence of events in that space, we might find it more difficult to imagine how we could be fooled by the coordinated stimuli of all or most of the senses acting concertedly during a period of purposeful interaction with our surroundings (or apparent surroundings!) Though I can't help thinking the boffins of "virtual reality" would rise to this challenge, I doubt that in the long run the "virtual" is eliminable from the worlds of their making. (Will the virtual grenade ever kill?) When we include in this vista the whole milieu of accumulated effects and expectations that contribute to our knowledge of the physical world, between womb and grave, nothing seems capable of shaking our faith in the exclusiveness of that category, even though from time to time we may err about what properly belongs in it.
6.6 Personal interactivity
It is, therefore, our personal appreciation of the nature of the subject matter of which experiential propositions speak which supplies the grounds for distinguishing objective from subjective types. How do we arrive at this understanding? Although the classical idea of public verification has little bearing on this central question, we can surely put into service the notion of immediate or short-term personal verification - a kind of almost instantaneous or ongoing accreditation of objectivity achieved by the cerebral integration of stimuli produced through the interaction of one's body with its environment. (In short, people know what they are doing, and that's enough!) This, however, is the fountain of our objective knowledge and not its ratification. Granted such a diminutive suggestion would be of little worth in celebrating the acquisition of scientific knowledge. But then philosophers of the twentieth century empirical tradition have fostered overly credulous sentiments concerning the relevance of science to the concept of existence and the formation of objective values - sentiments with which scientists themselves rarely consort.
Are explanations of this kind what we are really seeking? If people almost never report any difficulty in distinguishing personal experiences from objective events; if there just are these two camps lying on either side of the boundary between oneself and the world in which one lives, moves and communicates; and if all we need is a nudge now and again to correct our terminology when speaking of them; then perhaps philosophers have been asking all the wrong questions. Until someone with a more inventive mind can make us see the problem from a radically different perspective, I think we must content ourselves with the bald conclusions that there are material and mental phenomena and corresponding statements which refer to them; that the phenomena in the former category are distinguished from those in the latter primarily by our personal, often immediate, comprehension of their relationship to the environment with which we interact; and that the two kinds of proposition cannot be distinguished in terms of verifiability, fallibility, extensionality or public accountability. Ultimately the factors that count for most in communicating either kind of information are personal interrelationships founded on integrity and trust. Somewhere in this forsaken territory lurks the essence of reality, and finding it remains one of philosophy's most challenging quests.
7. SOLUBLE PROPOSITIONS - PURE LOGIC REJECTED
7.1 Interpretations of formal logic
The backbone of modern logic is an uninterpreted algebraic syntax, some of whose formulae are soluble, which is to say, in classical terms, that they are either tautological or inconsistent or that they can be reduced by a specified calculative procedure to one of the values 1 or 0. Logical atomism and positivism, with their notion that the variables - and only the variables - of this syntax can be replaced by unanalysable propositions representing basic facts in, or impressions of, the real world, have promoted the belief in a system of reasoning that retains its logical rigorousness even when interpreted. Thus it's fashionable to think of "ordinary" logic roughly as comprising:
1. unassailable rules of thought or canons of pure reason, faithfully represented by
2. symbolic expressions which expand "analytically" (or "constructively") from more or less classical axioms or in accordance with classical rules of computation, and
3. the logical features of whose formulae remain intact when appropriate meaningful expressions are uniformly substituted for variables of the formulae.
I shall refer to formal systems which operate in all three modes as formal but topic-neutral* (a term borrowed from Gilbert Ryle), reserving the names pure and systematic for systems whose interpretation (3) is supposed to be confined linguistically to (2) and ontologically to (1). Which is to say that the variables in a system of pure logic may be substituted only by appropriate well-formed expressions of the system itself, while the connectives are interpreted in overtly logical terms, implying, presumably, that they stand for the basic devices of reason and that we reason correctly only if the devices we use conform to the "analytic" conventions of the calculative structure. I shall use the term formalist for any kind of allegedly analytic, symbolic system which is either reckoned to be ontologically vacuous or whose "ontology" is confined to well-formed formulae of the system. In what follows we shall largely be concerned with questioning the worth of each of these three perspectives.
*It's arguable that pure logic is inherently non-topic-neutral for two reasons:
We shall assume, first, that we can think logically, regardless of the status or derivation of this talent. It is widely acknowledged, though - one might be forgiven for thinking - scarcely believed, that the syntax of classical logic is open to non-logical interpretations, or to no interpretation at all. The set of procedures which I often adopt to solve problems in formal logic, for example, treats standard propositional calculus as if it were a binary modular arithmetical algebra (see Endnote 3). One of the several advantages of this method is its psychological friendliness, in that the elementary arithmetical calculations encouraged by its symbology are more familiar than the Boolean-style calculations presented in most textbooks on logic. In doing this binary arithmetic one soon becomes aware that what one is doing is nothing but arithmetic and involves no trace of logic. Psychologically, one follows a calculative procedure which seems to be entirely devoid of any "argumentive" undercurrent, its utility consisting precisely in the ability to completely replace argumentive by arithmetical deliberations. And although one could develop such a system by the conspicuously logical technique of deduction from axioms, this seems a presumptuous and unnatural approach. The system evolves more spontaneously from a fairly arbitrary series of demonstrations, in which one learns to recognise the general features and behaviour of "well-formed formulae" - a strategy that relies on a less painfully acquired cultural repository than the comprehension of the concepts of lemma and deductive proof demanded by the standard approach to logic (which, alas, also depend on pattern recognition).
(i) If a logical formula (containing connectives) is substituted for a variable in a logical expression, a topic is introduced, namely, more of the same kind of logic.
(ii) If a logical expression is interpreted as, or used as, a guide to reasoning (i.e. #1 above), it's central topic is "reasoning": as opposed to being a bunch of uninterpreted or useless symbols, or symbols given another, non-logical interpretation. (More on this below).
But if a given structural scheme - even though apparently well suited to logic - is open to interpretations having no logical significance, we shall want to ask, first, whether there are any general principles for expanding, contracting and maintaining the integrity of the structure (transformation rules) and, if so, whether they are principles of "reason"; and, secondly, what is the relationship between the structure and any interpretation we choose to give it - in particular, the "logical" interpretation. Though for the most part beyond the scope of this article, the first question surely demands special attention; for here we have the chance to confront some principles of thought which are more fundamental than those of logic, and which have relevance both to computational reasoning and to our primary experience of the physical world - notions such as recurrence and similarity, recognition and expectation, differentiation and association, form and symmetry, abstraction and interpolation, substitution and extraction, and above all the difference between organisation and chaos. I take the view that the study and exposition of most, if not all, of these structural principles fall within the province of mathematics, that "logical" structures and "logical" reasoning form only a small part of this field and that mathematicians might do well to break their ties with the traditional logic-based methods of doing maths. As an analytic discipline, however, mathematics in all its aspects must eventually endure the same censure as logic (see §8).
It is the second question that concerns us now. The symbolic expressions employed by logicians comprise only patterns and sequences of marks and evidently have no intrinsic logical import. A formula or finite sequence of formulae which possesses a certain kind of symmetry (as determined by proof, for example) may properly be said to be correct ("orthologous" in my terminology), but only to depict logical validity. It's difficult to see how such representations in themselves might be considered fundamentally logical. This remains true even if we suppose that the symbols dictate their own functional possibilities, the way they are to be used, the rules for manipulating them - which, after all, merely chase the patterns that can be constructed from them. The same rules apply, the same possibilities for pattern-building occur, whether or not we interpret these constructions "logically". Even so, it might be argued that this symbology and these rules do accurately reflect whatever it is we are doing when we do logic; or that they faithfully represent a non-mental metaphysical realm of logic that is somehow distinct from these underlying patterns. For a moment let us admit this possibility, viz there is an activity or realm of formal, topic-neutral logic whose properties can be faithfully represented by symbolic structures developed by logicians.
7.2 Logic, meaning and relevance
In §3 I posed the question: if "The cat is in the cupboard" and "The cat is not in the cupboard" are both observationally testable propositions*, how can it also be maintained that these very same two propositions are logically contradictory? [*Actually I tentatively proposed that this pair contravenes the general constitutive proposition "No organic individual can be in two places at the same time".] There are a number of ways of visualising, and perhaps attempting to resolve, this quandary. One approach is to deny that there is any connection between the propositions as such and the abstract treatment they receive in systematic arguments. We simply replace one proposition by p and the other by ¬p, where "¬" is a strictly logical operator. But then of course we fail to mark any relation whatever between the original material propositions, much less to offer any warranty that the facts will comply with the formal design invoked to speak about them. If this position is extended consistently to all argumentation, the queer consequence appears to be that rational discourse has nothing to say about reality. For how can logic both maintain its regal aloofness and apply to practical situations? - as if there were an infallible structure of pure reason, on the one hand, and a less than perfect real structure to which it corresponds, on the other. But supposing, regardless, that someone had reason to subscribe to this view: one of the principles they would seem to be endorsing is that there is a pure logic comprising formal relations, though the related terms be meaningless.
Alternatively we could interpret the p's and q's of logical formulae empirically while giving the connectives (including the one-place connective, "¬") systematic interpretations. This stance, in which the bones of logic retain their respective bits of flesh, so to speak, seems to be especially prevalent despite the philosophers' recognition of its shortcomings. In particular, one faces the practical difficulty of how to work with the idea that the basic operator "¬" remains purely logical in interpreted propositions of the type ¬p while the contained proposition p is experiential. This leads to hopeless tangles. "The door is open" would have to be translated as "It's not the case that the door is closed" (or vice versa, but not both). We should have to deny that we could investigate both whether the door is open and whether it's closed. We could not reasonably substitute unnegated terms for negated ones in arguments. Clearly the notion that a proposition of the form p or ¬p is analytic, though p be experiential, is nonsensical.
A step forward is to exclude "basic" uses of negation from logic. On this account truth and falsehood in relation to simple variables are regarded as empirical notions, ¬p is interpreted as "p is false" and so negation in this situation has an empirical meaning. All other connectives, however, are regarded as logical, and expressions such as p & ¬p are regarded as representing analytic inconsistencies. But this only moves the difficulties a little further up the line, in addition creating incurable problems with the interpretation of negation in complex formulae and at the same time accentuating the underlying conceptual dilemma with all these approaches, which is as follows.
Given that if "p is true" and "p is false" both express experiential propositions they cannot also be logically contradictory, it would be capricious to ignore this by positing a formal relation of contradiction between them. Furthermore this care must be extended to all pairs of apparently contradictory basic propositions whose truth is judged by meaning criteria other than those of pure logic. Only after eliminating all meaning and substituting a systematic notion of truth might we become entitled to engage in systematic argument. But now supposing we do eliminate all meaning, as suggested in the first approach. It is one thing to posit, as a formality, that two propositions be contradictory, quite another to appreciate that they really are so: only if we understand the propositions can we then understand how they conflict with one another. Throw out the meaning and the possibility of contradiction goes with it.
The concept of contradiction - one of the most fundamental in logic, along with identity - thus relies on the interdependency of meaning between the two opposing terms. It is surely just as obvious that no relationship among experiential propositions carries anything like the force of logic unless those propositions are not only meaningful, but meaningful in a way that makes them relevant to one another. Without mutual relevance there can be no "logic". In particular, no argument is valid unless its terms are relevant both to one another and to the purportment of the argument. I'll try to explain this principle only in relation to material propositions (similar considerations apply to propositions of other types).
There are of course numerous ways in which material propositions may be mutually relevant. Most philosophers appear to acknowledge the indispensability of the notion of relevance in "law-like" conditionals, including not only causal, subjunctive and counterfactual conditionals but all conditionals which contain any trace of a law-like connection, and which, therefore, one can at least find excuses for declining to call "truth-functional". And while most philosophers would also admit that there are also molecular propositions of non-conditional form which are incapable of a systematic interpretation, I doubt whether many would find this weakness in all material propositions. It seems to me, however, that "relevance" is a general prerequisite for valid argument, being primarily a condition among propositions that attends the rejection of logical atomism and the notion of pure logical relations that goes along with it. Formal relations are displaced by experiential relations. In contrast, classical logic treats "material" relations (in the sense of "material" implication) as if all propositions are experientially independent, failing to recognise that the semantic continuity that underpins the fundamental concepts of identity and contradiction extends to all other relations of experiential propositions.
No one can doubt that when a proposition such as "Had you doused the fire nobody would have been harmed" is represented as a conditional (p
q) it is non-truth-functional, the truth of the whole relying upon a certain kind of dependency of meaning between antecedent and consequent. But what of "John Howard is Prime Minister of Australia and the moon is made of marshmallow"? This might well be taken to be a conjunction of two physically independent propositions which happens to be false as a whole just because one of the conjuncts is false. For, although there may be both philosophers and politicians who might think otherwise, the composition of the moon appears to have no bearing on the fact that John Howard is Prime Minister of Australia. But if these propositions are really unrelated we might well ask: why would we ever want to, and how could we, include them in the same argument, and how could we allot truth values to the propositions, on a par with one another, as it were? In what circumstances would we unflinchingly call the one proposition true and the other false, without changing pace or perspective?
In any ordinary conversation about the status of John Howard we would indeed have to alter our perspective if the subject of the moon's composition was abruptly introduced. But academic conversation tends to be unordinary, taking into its fold everything that exists or might exist anywhere and anywhen, and freely dishing out truth values all from the one pot. Not surprisingly, when it invents sentences the interconnectedness of whose components is obscure, it finds instead only paradox. The present example is surely not that "far out": both its conjuncts relate to our planetary system in 1997, and both represent facts which are investigable by the resourceful observer. Here the academic's concern just marks the perimeter of the domain in which the clauses are materially relevant to one another. As practical conversationalists, on the other hand, we are interested in whether both clauses are relevant to any useful argument (and, usually, to some specific argument). For I take it that no one except the odd philosopher or astrologer would ever want to argue empirically about propositions whose material connections are so remote. Should it happen that we do decide to talk about the composition of the moon, the Prime Minister's eyebrows and the price of chips in Moscow all in one breath, then hopefully we shall be aware of the limited nature of the association between these topics and not find paradox where really there is none. At the same time we shall appreciate that we cannot talk about such matters at all unless we recognise some connectedness of meaning - an interdependency that is sufficient unto the argument, so to speak.
This, then, is the dilemma: propositions are either meaningful and not formally related or else meaningless in which case it might be presumed that they could be formally related. But if they are meaningless it becomes impossible to see what is involved in such a relationship: meaningfulness is required for relationships such as contradiction to be comprehensible. Therefore either we must be "purists-gone-bust" and deny that there are logical relations, or, as ascendant pragmatists, admit the use of the word "logical" on the understanding that such relations draw their life from the meanings of the related terms and not from a conspiracy among mathematicians.
Because relevancy comes in a multitude of colours, there's no place for a unique mode of reasoning that is independent of the subject matter with which it deals. Take away relevancy and the relevant logic goes with it - there's no such being as a pure, topic-trading but topic-neutral logic. On the contrary, far from being topic-neutral, logic appears to be remarkably topic-dedicated. This does not of course imply that we cannot assort logics, in a rough and ready manner, into general classes appropriate for different types of purpose. It is one thing to abstract from uses of reason, quite another to install a formal procedure that regulates reason. We can recognise chairs and tables and use them effectively without appointing a warden to control their use (as if there were not enough to go around).
7.3 Systematic logic
Although it may be conceded that when logic has to deal with the inexactitudes of the real world it must follow a muddy, twisting track, still one might imagine that there is a pure, analytic logic which consists in the application of typically logical reasoning to the constructions of mathematical logic. This seems plausible because it's possible to substitute the variables in a logical system by well-formed expressions of the system itself, thereby endowing them (so one might think) with sufficient meaning to preserve the quintessentially logical features of logic, such as contradiction and deducibility, which apparently ensue from the use of appropriate syntax. But as intimated at the beginning of this section, this ploy alone does not genuinely fulfil the semantic requirement for systematic formulae to be interpreted as "logical". For it is impossible to eliminate the p's and q's (which are unanalysable and therefore meaningless) from molecular propositions. Replacing them with more complex formulae doesn't help, because when the proposition is reduced to normal form meaningless terms remain as basic components, so the logic as such disappears. And even if it could somehow be supposed that every variable tacitly represents a structural expression, this would bring us no closer to justifying our belief in a logical interpretation.
Consequently a semantical idea of logical relations is a prerequisite for beginning logic. I don't know what the exact nature of this idea would be, but we might imagine that it is an abstraction from the material relations of which we have been speaking (rather as the concept "jump" might be regarded as an abstraction from a large number of instances of jumping). The only hope of reforming this street-wise generalisation lies in persuading it to follow unswervingly the path dictated by an axiomatic structure. Provided our logic conforms rigidly to a mathematical pattern, will it not then be analytic? Wishful thinking indeed! The question is grounded in pure empiricism and presents no genuine challenge to any of our previous positions.
7.4 Formalist logic
I therefore conclude that the three aspects of logic with which we began cannot be combined and that logic as an argumentive process is thoroughly empirical. However, there are formalist systems employed by logicians in the study of their subject, and by others as aids to thinking. The well-formed formulae of these systems belong in the general category of structural or mathematical propositions, whose calculative properties appear to provide some of them with the credentials for analyticity. Soluble propositions are one such kind (see §9). I now wish to argue, by example rather than by principle, that not even these formalist structures (or those of them which are supposed to qualify) are analytic. The symbolic constructs of mathematicians are themselves "impure", in as much as they are restricted by the same experiential conditions that shape every other aspect of our lives. Far from dealing with the indubitable, mathematicians inhabit a fairyland whose images are frequently as nebulous as any in the most recondite of metaphysical systems. (Not that mathematics is any worse off for this, as we shall see.)
In attempting to promote this evaluation, instead of logic I've chosen to consider arithmetic, in particular some aspects of the concept of number. Like symbolic logic, arithmetic is essentially formalist and calculative, but unlike symbolic logic it doesn't scream out for interpretation beyond its own symbology: it deals primarily with the element of recurrence within a calculative system. It therefore lends itself to a more incisive impression of syntactic structures and their empirical constraints. At the same time, I hope that, by applying a little pressure to the borders of normality, we shall uncover something of the nature of arithmetic itself.
8. ARITHMETIC AND STRUCTURE
Arithmetic, like geometry, is a curiously mixed pursuit. Few would dispute that a proposition such as "There are about 20,000,000 people living in Australia" is entirely empirical, because (provided one does not think too hard about it!) it contains no trace of the calculative procedures that characterise the science of mathematics. Number, as used in this proposition, is apparently a purely empirical concept. As soon as one begins to calculate, however, the character of number seems to change. That the population of Australia is close to 20,000,000 is a fact, but that 400 x 50,000 is exactly equal to 20,000,000 is a fact of a very different kind. Propositions of the second kind have long been considered analytic, in the strongest sense of that word. In fact it has sometimes been held that arithmetic tows the analytic line, as it were - it is the archetypal system of analytic propositions. While this may be going too far, it is apparent, at least, that the philosophers of mathematics and logic have done their best to divest number of its empirical connections and transform arithmetic - indeed the whole of mathematics - into an all-embracing analytic discipline. In spite of some devastating snags identified by Russel, Godel and others, this is pretty much how maths is envisaged by most people today. Yet many of the foundational concepts of maths are blatantly empirical, while many of its fundamental axioms (such as that every number has a successor) seem to beg the question of empirical status. My aim in what follows is to give credence to the opinion that, even in its most formal aspects, arithmetic cannot throw off its empirical chains.
A problem for many philosophers, but few mathematicians, with modern arithmetic is the riddle of the meaning of infinite numbers. In fact Georg Cantor, one of the founders of number theory, divined that there's an infinite hierarchy of infinite sets, but, thank goodness, we shall barely need to touch one of them. While of course I don't question the indispensability of the systematic concept of infinities to mathematics, I do subscribe to a certain intuitionist/formalist precept which grants priority to finitary concepts and syntax, crediting them with a type of meaning or "contentfulness" (David Hilbert's term) which is absent from notions of the infinite and from incompletable expressions. The thrust of my concern, however, is with the concepts of the finite and the denumerable, not with the infinite. It seems to me that two grey areas have been neglected, lying between the infinite and the observably finite or numerically accountable - one of them at the immensely extensive or multitudinous end of the scale, the other at the minutely divisible or infinitesimal end. It is here that I believe mathematics fails, that its failure is in principle demonstrable and that its weakness in this area impinges upon the whole of mathematics and rational thinking.
Although it is now well established that mathematics is "weak" in the sense that it contains uneliminable elements of randomness, uncomputability, paradox and unprovability, I don't know whether these failings are related to those which we are about to consider. I suspect that, despite their acknowledgment of the former kinds of weakness, many mathematicians would reject the notion of empirical conformity, and would doubtless think my conjectures are intuitionism-gone-crazy (although, so far as it goes, the "theory" is essentially realist). Should they remain of that opinion, however, they will one day be jolted into submitting alternative theories when the inevitable happens: mathematics will become so far stretched that it will cease to produce consistent results for reasons which seem non-systematic. While doubtless much of my story is old hat, I'll try to arouse a glimmer of interest by adopting a speculative but graphic approach. In offering some rather secular and simplistic illustrations, I hope to supply clues as to how it might be possible to test the central idea; for it must be assumed that this is either a cosmological theory with practical consequences or else metaphysical nonsense.
8.1 Space as structure
First, a little speculation about the nature of space is in order. When I say "space", I mean local or domestic space, the space that we live in and about which you and I, as well as the Euclids and the Newtons, tend to form intuitive, pragmatic judgments. It is widely appreciated, I think, that the impressions of space received through the various sense organs are entirely different in kind from one another: there's no resemblance between visual space, auditory space and tactile space, and no inherent reason why one should suppose any object in a visual field to be identical with one perceived via any other sensory route. What, then, is the binding force that unites and coordinates these different kinds of sense data, along with their peculiar space-like perspectives, causing us to embrace them as representations of a single space? A very plausible answer is that they are related by a mathematical or logical structure. But once having brought ourselves around to that outlook, it begs little further insight to reach the judgment that space consists of nothing more than a mathematical structure. For nothing is required besides mathematics to unify these different sensory perspectives; the assumption that there is a "noumenal" physical space apart from pure structure is metaphysical and needless. Space is nothing but mathematics.
Since the human brain is a structure in space, and has evolved primarily as a mechanism for sustaining itself in its spatial environment, one might suppose that the conditions of pragmatic thought in general, and of logical and mathematical thought in particular, are themselves influenced by the nature of space. And unless there are aspects of thought which are independent of the physical existence of the brain, conceived as a spatial object, it would be very surprising indeed if the topology of human thought turned out to be unrelated to that of space. This is of course a chicken-and-egg situation, but one that is of no immediate consequence: the significant idea is just that local space and the way we think have a predominantly common structure. (Admittedly the equation must be extended to include at least space, time and force in a broadly Newtonian structure, but the resultant complications would not further our cause.) The crucial point to grasp is that there's a bigger chicken and a bigger egg: the logical and mathematical scope of the human mind is subject to the very same constraints that the mind attributes to the local universe. Conversely, the local structure of the universe as comprehended by the human mind is limited by the same intrinsic topological factors that determine the kind of logic and mathematics of which human thought is capable. Mathematics, the mind and the local universe have a common foundation.
The reality of the situation, however, is that the physical universe does not (according to present-day reckoning) possess the intuitive Euclidean/Newtonian structure we once believed in. It's different, and therefore one should expect the foundations of mathematics to differ in a related way, or to collapse in certain circumstances. The situation is evidently complicated by the fact that the mathematics we have available to undertake cosmological enquiries is just the mathematics whose foundations are in question: it seems we need to understand the nature of mathematics before we can describe the universe, but there's a vicious circle involved. Nevertheless, I believe the hypothesis that the foundations of mathematics are awry is in principle testable, and, if true, there are many far-reaching consequences, some of which will be noted in the Postscript.
8.2 Formalist arithmetic
There are, of course, countless "theories" about the nature of arithmetic, ranging from the purest formalism through various psychologistic interpretations to the most implacable realism. While my own view leans toward formalism, this is not so much a philosophy of arithmetic as a judgment that all calculative symbolic systems are intrinsically arithmetical. Arithmetic is the science of recurrence. But while the notion of recurrence is crucial to understanding symbolic systems it applies well beyond that sphere, being an integral component of almost every aspect of human experience. Consequently this predilection for formalism is of little significance.
Furthermore the question of the ontology of arithmetic is of no immediate importance. For there could surely be an agenda - call it "arithmetic" or not - which is formalist, as well as one in which the symbols are objectively or psychologistically interpreted (so that the meaning of the symbols is not just more symbols in the same system). In respect of the latter discipline, however, one might expect to find a variety of interpretations of "arithmetic-like" syntax and hopefully a reasonable explanation as to why any of them be regarded as distinctively mathematical; in particular one would feel entitled to an explanation of how the analytic character of such a discipline is conserved in its ontology rather than merely in its syntax. Regardless, I shall initially assume that arithmetic is formalist (i.e. it comprises nothing but symbols - I don't think this coincides with formalism in the modern sense, which puts more emphasis on meaninglessness and rigid proofs), so that our discussion can easily be related to the question of "pure logical syntax" which is our primary interest.
Within a broadly formalist framework one can still approach a number of issues from different angles: specifically, it looks as if one can adopt attitudes which might appear, to the sophisticated, to be ontologically different. For example, one might take the view that arithmetical syntax comprises nothing but transformations of utterly meaningless symbols; or one might say that the system contains names or tokens which denote other systematic objects (which in turn can be used as names or tokens), thereby giving the impression that the syntax is after all meaningful. Although this particular distinction (which may be verbal only) doesn't affect my case, I prefer the second approach because it buoys my view that arithmetic is best envisaged as possessing a "fluid hierarchical" structure (see below).
8.3 Number and tokens
Arithmetic begins at the small end, the human end of the mathematical spectrum. It begins with counting. This truism (the cradle in which, it might be thought, the avowed intuitionist chooses to spend his life), seems to have been respected by the inventors of number theory, notably Cantor and Giuseppe Peano. And how could things have been otherwise? Who would have taken any notice if the theory had had no footing in the nuts-and-bolts concept of natural number with which we are all familiar? Are we not entitled to assume that this is what the theory is about? Unfortunately when arithmetic is stretched into the realms of the uncountable, we can no longer take this assumption for granted.
I know I'm not alone in harbouring the feeling of being duped by Cantor's theory of sets and numbers, the feeling that it's somehow circular and fails to get to grips with number as such. (This is not a psychological problem, but rather a problem about the validity of real-world proofs for mathematics and, conversely, of the existential status of the mathematical objects so defined.) Although Cantor himself was obviously no formalist, his explanations are uncompromisingly syntactical in as much as they depend on establishing one-to-one correspondence relationships between series of numeral-tokens, that is, between symbols occupying a small zone of Euclidean space. Its assumption that these series of demonstrations are indefinitely extendable and/or indefinitely interpolatable appears to rely on the assumed topological properties of infinite extension and infinite divisibility of the space in which they are represented. And although, under a less formalist interpretation, it might be held that these representations do not depend on real space for their actualisation, it would still appear that whatever it is that's supposed to be going on requires a logical space with similar properties. For without this assumed space, one could never predict that every supposed one-to-one correspondence would in fact be unique, or even possible, when the series is "represented at length". Thus Cantor's definitions of numerical infinities rely on an undefined notion of spatial or logical infinity. The acceptance of his technique as a valid mathematical method depends on the unfounded and improbable belief that what can be physically demonstrated on a piece of paper can be extrapolated ad infinitum to increasingly unwieldy gesticulations that cannot actually be symbolised anywhere or anyhow.
As Cantor draws upon ever more picturesque techniques, the limitations of the page become increasingly bothersome and the proofs less convincing. For example, in the procedure that's supposed to show that the number of proper fractions (rationals) is the same as the number of cardinals, Cantor introduces two complications. First, in order to deal with the the fact that every fraction can be represented in an infinite number of ways, he has to delete an infinitude of irrelevant fractions (namely, all those whose numerator and denominator have a common factor). Secondly, he has to coax us along a zigzagging path through his two-dimensional array of fractions, skipping the irrelevant ones on the way. This is an extremely "spatial", undependable looking procedure (see Endnote 5).
There is of course no difficulty with the notion of correspondence of relatively small, finite sets which can in fact be matched and whose number might feasibly be ascertained. But the extrapolation of the notion to larger sets involves the use of synoptic tokens which do not themselves possess the properties of the sets referred to. This, however, is a feature of arithmetical syntax in general (irrespective of ontological presuppositions).
Yes, arithmetic does begin at the human end of the mathematical spectrum, with counting. Formalist accounts may accommodate this maxim by recognising that some expressions in arithmetic are relatively "primitive", while others are really names or tokens for (often exceedingly extensive or infinite) collections of primitive symbols. In other words, there are synoptic tokens whose meanings are complexes of more primitive signs, and which in principle can be expanded analytically, by correct calculation, into the complexes that they represent. Naturally there are degrees of primitiveness, degrees of complexity and little inclination to single out any particular set of signs as being "the meaning" of any other set. But one can surely sympathise with the idea that a token such as 123 can be expanded to one of the form 1+1+1+....+1 which better captures the literal meaning of the original token in so far as it contains just as many 1's as the number signified by 123: it exemplifies the number and does not merely signify it. But now what of the token 219937-1? Evidently this expression could not be expanded to one of exemplificatory form (1+1+1+....+1) even though one had begun to compute it at the beginning of time using every available minuscule in the universe. Yet more than twenty-five years ago this number was proven to be a prime, and very much larger primes have since been discovered. (And of course still larger numbers can be expressed in token form. For example, there's a number called a moser which is unimaginably huge, but which is easily defined using only the number 2, a few geometrical symbols and the concept of exponent.) So, on this view, a proposition such as 101000 = 10500 x 10500 is analytic but meaningless because the most primitive forms of expression betokened by the terms of the equation cannot in fact be completed. But regardless of whether some signs are more basic than others, it remains true that arithmetic alludes to some translations of signs which can never be depicted or used because they are too extensive or, rather, their components are too numerous! Since no such translations can possibly exist, there is strong justification for the claim that any reference to them is "uncontentful".
If this is right, then the sign 219937-1 cannot be used to refer to an expansion of the form 1+1+1+....+1 and remain a bona fide component of an analytic system. Owing to its central concern with the rudimentary concept of number, however, conventional arithmetic does contain references to uncompletable signs. Consequently arithmetic as a whole lacks the credentials for analyticity. And (for those whose concept of number is not tied to mere symbols) it's clear also that no token in an arithmetical system can denote an extrasystematic occurrence of a precise number (such as a counting of objects) if such an occurrence does not, nor ever could, exist.
Thus arithmetic contains expressions denoting either incomplete (and uncompletable) tokens or uncountable sets of objects, or both. Although, so far as I know, this verdict doesn't at present detract from the utility of the analytic craft of sign juggling, it might be prudent to keep in mind both that sign juggling is not necessarily the same as number crunching and that the analyticity of mathematics is vulnerable just to the degree that it projects its language beyond the reaches of the conceivable. Much as Newtonian physics is vulnerable to the degree that its domestic language of space and time loses meaning when we want to converse with the electrons and the stars. (Is it simply the physics or are there already signs of the maths going wrong?) As with other sciences, mathematics holds no built-in guarantees of performance.
I have used a spatial picture to show that we cannot predict how numerical signs behave when we try to imagine an extrapolation of a series into regions beyond the immediate environment which establishes the conditions of sign writing. It cannot be assumed that the framework and assumptions applicable to manageable numbers has legitimacy for gigantic numbers. Had we been more adventurous and chosen an illustration befitting our times, such as the way that computers handle arithmetical information (as has been done, for example, by Rolf Landauer, 1986), we should have arrived at just the same conclusion - that arithmetic is empirically constrained. A more anthropocentric illustration, however, is provided by the concept of counting.
Numbers begin with counting! According to the axioms of natural number ascribed to Peano (1908), every number, n, has a unique successor, n+. An intuitionist might take this to mean: take any particular number, there is just one number that is one greater than it. But how would you in practice take any particular number? Suppose it was a very large number whose value could not feasibly be checked by counting. How would you then know that you had the number you intended to take? The notion of identifying an uncountable number as being a particular number is incomprehensible. On this account, Peano's axiom seems meaningless because it doesn't satisfy the criterion of real countability.
If counting is to be explained in terms of putting signs and objects into one-to-one correspondence with one another, then nothing more need be said: we have seen, first, that this is an empirical procedure and, secondly, that it cannot actually be done with very large numbers. But is it not also possible to count by rote, as school children often do, by learning the sequence of signs without attaching them to physical objects? Well, how can we distinguish counting by rote from counting things? The first kind of counting seems to consist only in reproducing the conventional tokens for successive numbers in the series of ordinals, while the second kind involves both reproducing those tokens and placing them in one-to-one correspondence with the members of a set of objects. It seems to me, however, that the distinction lacks substance: when counting by rote, we do in fact put different signs into correspondence with instances of something, even if only intervals of time. Of course, if we are counting events, or just counting off definite intervals of time, such as seconds, it's easy to contend that we are employing the correspondence procedure. But it might seem that counting by rote lacks objectivity, that it doesn't involve events and that the time intervals are somehow too arbitrary and inconsequential. I doubt this: the activity of counting itself supplies the events and thereby demarcates and orders the intervals.
One possible objection is that we cannot in principle do a recount: objects can be recounted, events can be recorded and recounted, but how does one recapture a counting per se? Well, couldn't we replay a recording of our counting, assuming we counted aloud, and count our counting again, so to speak? Suppose we just recount the noises as such, without taking notice of their form. Then surely we could be said to be recounting our original count - which was, so to speak, a labelling of noises contrived by giving a particular shape to each noise. I think the following consideration completely justifies this view. If we count by rote, say, from 1 to 20, it would be in order for anyone to ask if we have counted right. If we did not count right, then at some stage in our counting the number of noises delivered up to that stage would not correspond to the meaning of the noise uttered at that stage. (I say "at some stage", not necessarily upon reaching 20, for we could have made two or more mistakes which cancelled each other out, and so have made 20 noises yet not have counted right.) Now suppose there was in principle no way of checking the count. In what sense could we then be said to be counting at all? How could we ever be sure that we were not simply uttering noises at random? Counting by rote entails counting objects, namely the signs that constitute the counting; the signs are labelled by giving each of them an unique, conventional form; if the counting is right, the form of every relevant token corresponds to the number of tokens delivered up to that stage. Counting cannot take place in a void. Even when "nothing is being counted", counting is an obstinately experiential process, temporal and psychological. And since every sign that represents a natural number must represent a countable number, or else be meaningless, arithmetic in general is empirical. Its applicability to the real world is irrelevant to this argument. Arithmetic is inherently real.
If counting is experiential, the proposition that every number is countable in principle is meaningless: something, even if only the numeral tokens themselves, must be countable in practice. A numeral token such as 101000 in isolation cannot stand on its own feet - we cannot tell whether it stands for "the number" it's supposed to, nor conceive of its basic meaning, nor ascertain whether any such number exists, since it is not and never will be literally countable even on the fastest computer that could in principle be designed. On the other hand, the number 1000 is meaningful because it's countable in practice and there are sets of objects or events that can be placed in one-to-one correspondence with the series of natural numbers up to and including 1000.
How else might we attempt to count things? Although it isn't necessary to literally count the members of sets to compare their number, some method of matching them is required. It's easy to show, however, that any procedure for matching sets or patterns at some stage involves at least as many discrete operations as there are objects common to both sets (or elements common to patterns), and therefore a similar number of operations as would be involved in doing a literal count. The "contentfulness" of number depends on this pragmatic potentiality and cannot be captured by shortcut techniques. It's of little account whether we imagine these operations to be essentially spatial, temporal, psychological or belonging in some more abstruse logical space; it matters little in what framework we conceive of the existence of numbers. Given a coherent view of space, time and "psychological space", we shall find that the various formalist and psychologistic concepts of number are practically identical. Somewhere along the line we turned the manuscript upside down: number itself calls the tune, erratic though it be, and both space and time dance to its strange music.
8.5 Mathematics and meaning
Is there a unique category of mathematical, as opposed to structural, propositions? The breadth and complexity of the subject-matter of mathematics make this extremely unlikely. Without getting involved in questions about the nature of mind and artificial intelligence, one might presume that an effective test of whether mathematics comprises only pure structure or demands a psychological interpretation is whether computers can do it. But the test is marred by the need to resolve, in turn, what it is that computers do, and (having decided that) by the purely terminological question of whether it's proper to call what they do "mathematics". In regard to classical logic, for example, computers can perform the appropriate systematic operations, but I doubt whether they can do logic. I don't hold quite the same doubts about their capacity to do mathematics, or some mathematics, anyway.
Much depends on what one makes of the business of interchanging symbols that have complicated meanings. For while the principal domain of mathematics is structure of all kinds, it seems to me that as a formal system it comprises nothing but a sign language, employing signs and about signs. Its alleged analyticity consists in its calculative procedures for translating signs into other signs. Mathematics is everywhere hierarchical, it employs an indefinite series of types. A clear-cut distinction between mathematical and metamathematical levels is nowhere to be found, but the difference invades mathematical systems in subtle ways at every point. Signs (tokens) are objects and vice-versa: the objects that are the meanings of signs are themselves signs: the distinction is relative. Essentially there are symbols that are more synoptic and symbols that are more expansive (such as strings of operations or sets characterised denotatively). But the analytic nature of mathematics becomes suspect whenever a sign cannot be completed, or when the objects in a set that a sign stands for cannot be listed, or when the operations required to expand an expression cannot be enumerated. One can only say that an expression (such as that 219937-1 is a prime) is meaningful if one can envisage a practical, "uncompressed" procedure employing a finite number of steps to prove it. And then it is meaningful just "to the extent of the procedure" and not in any sense that exceeds it. Thus the number of operations required to disclose the primitive or expansive meaning of an expression is of paramount importance. Number is of the essence.
Mathematics in the large therefore differs little from physics: it has inherent experiential limits and it is subject to similar conditions of uncertainty and relativity. Expressions with colossal or infinitesimal denotation, however, might be amenable to statistical treatment, so that mathematics could in some measure solve its own problems by adopting the same kinds of procedures for itself as it does for problems of physics. But I believe also that mathematics will turn out to be "assessor-relative" or subject-dependent. No procedures exist that will equally well serve every one in every time, place and condition. There is no almighty algebraist, no mystical realm embracing every possible mathematical construction. On the contrary, all mathematical structures, no matter how complex or inevitable - or how paradoxical or uncomputable! - they may seem, come into existence as the creations of calculating people, and all mathematical solutions must be understood by them. At the same time, this seemingly constructivist activity possesses an objectivity that is scarcely distinguishable from physical reality; and I can see little value in the belief that mathematics is endowed with a special sort of reality of its own.
I hope that now no difficulty will be encountered in transferring the example of arithmetic to logical syntax and similar constructions. All formalist systems operate within similar experiential constraints, and although undoubtedly the numerical aspect of symbolic structure is not the only one calling for consideration, like arithmetic, other formalist systems embody recurrence and a hierarchical symbolic structure comprising a gradation of signs from more primitive to more synoptic. Of any such system it can be said that meaninglessness and uncertainty grow as we attempt to extrapolate to increasingly remote cases a prescription that executes successfully within the local environment in which it was designed. Repeatability, "correct calculability" cannot be guaranteed.
8.6 Logic as inductive thinking
The position with systematic logic, then, is as follows. The uninterpreted syntax of classical (or any other) formal "logic" is but one kind of mathematical structure or calculative system. As soon as we attempt to enliven the syntax with the breath of logic, with the potential for genuine argument, we face a dilemma. Any instillation of meaning into argument blocks the possibility of pure logic, while the absence of meaning renders logic totally redundant: any calculative interpretation - or none - will then suffice. So for any argument one can say: either one is thinking logically but one can never know whether the logic will work, or one is performing a calculation but it doesn't matter what one is thinking; the computations can be executed mechanically. What one cannot do is either think or perform "analytically" - for neither reason nor symbology can engage the required infallibility.
What can it mean to be thinking logically when the paradigm - pure syntax - is devoid of logic? Well, clearly the source of logic is inductive thought, not deductive. It springs from expectation, our continual confrontation with the recurrent associations inherent in organic processes. And I do not believe any thought process can have greater strength than inductive strength. There can be no seventh category dealing in pure logic because there are no propositions of that sort. Of course, the reduction of symbolic patterns to forms that permit their status to be assessed requires some kind of intellectual effort. And although, as I pointed out, this need not be a logical chore at all, one would very likely opt for a procedure which one believed to be logical. What is it that one does in using this option? One simply draws upon the inductive mode of reasoning* one uses in solving many a real-world problem (consciously or unconsciously) and applies it to this particular practical problem. Doing symbolic logic per se, or any other "abstract" calculation, is just another empirical problem to work out, and like every other problem it is subject to experiential restrictions. (* I refer to induction as a mode of reasoning reluctantly. Induction does not involve any kind of straight-jacket reasoning at all, but combines an appreciation of continuity and regularity in nature with a feel for the relevance of events and situations drawn from all quarters of one's experience. So far, attempts to explain and formalise induction have all been futile, most of them pitiable. But I am claiming that one kind of reasoning is to be explained in terms of another which apparently has few of the characteristics normally associated with reason. Either we must understand "reason" more broadly or give it another name - Wisdom? Intuition? Common sense? Induction, understood as reasoning from the specific to the universal, won't do because (a) it begs the question and (b) the universal is mythical.)
The invention of mathematical logic has set all literate people thinking in reverse gear, and, because the authority of a respectable education has etched this discipline deep in their minds, it's going to be difficult to get them to drive forwards again. They have it in their heads that there is a pure logic which is somehow applied to objective facts, whereas the truth is that reason merely reflects the possibilities inherent in real-world situations, whether natural or engineered: we think correctly if we perceive these possibilities clearly. There is no formal logic - the logic of any situation emanates from the situation itself and the way it can develop in its environment, as seen through the eyes of survivors. So if we are to retain the name "logic" for the kind of rational thought that embodies inference and the sorts of linguistic devices associated with the classical discipline, we must say, not that it's a foolproof analytic technique, but that it reflects the relationships among actualities and possibilities in the real world.
Their trust in the necessity of mathematical logic has made it especially difficult for some philosophers to come to terms with relativity and quantum theory and some of their more exotic implications. As well, they have failed to understand and promote the thoroughly existential nature of the logics underlying both the old and the new sciences. This is a pity, because the scientific developments of the twentieth century provide a clear omen that the children of the twenty-first will succeed in escaping the shell of conventional existence which has been the prison, the fortress and the home of their ancestors for many generations past.