NOTES ON ANALYSIS
Translation © George MacDonald Ross, 1974–1999
1. Untitled, undated note (Couturat. pp.167ff.)
[167] I discover two parts to the Art of Discovery (inveniendi): Combinatoric and Analytic. Combinatoric consists in the art of discovering questions, and Analytic in the art of discovering solutions to questions. However, it often happens that the solutions to certain questions involve more Combinatoric than Analytic, as when one is trying to find a way of inventing (efficiendi) something in technology or in society; for in such cases one will have to seek means which are outside the thing itself. But in general, finding questions belongs more to Combinatoric, and solving them belongs more to Analytic. But there are two sorts of questions: (1) when what is sought is a future or past means of discovering or inventing something; and (2) when what is sought is a proof of the truth or efficacy of things discovered or invented by others. There is as much difference between these two as there is between the art of writing or speaking well, and that of passing sound judgment on what has been written. The testing of what has been discovered is purely analytic; but the art of discovery or invention itself is more combinatory.
On the other hand, these can be distinguished more precisely. Strictly speaking, Analytic is an investigation in which we separate the object itself into parts as exactly as we can, scrupulously observing the position, connection and form of the parts, and of the parts of the parts. Synthetic (i.e. Combinatoric) is when we add something external to the object in order to explain the object. E.g. dissection of animals is analytic; but suffocating animals in a vacuum pump and then cutting them up, is combinatoric. Examining liquids by distillation is analytic; but doing so by inducing a different reaction through addition of different liquids or powers is combinatoric. You will object that the fire for distilling, and the knife for cutting are brought in from outside. This is true, I admit; and it was certainly a combinatory operation that was performed by the first person to teach the art of cutting with a knife or of vaporising liquids by heating. But now that the use of such equipment is common knowledge, it should be as if the fire were permanently linked to the liquid, and the knife fastened to the corpse, since the idea of the one always conjures up the idea of the other. This is because we nowadays so often experience [168] those two things conjoined as a result of human intervention. Consequently, with the passage of time, certain operations which were once combinatorial will become analytic, after everyone has become familiar with my method of combination, which is within the grasp of even the dullest. This is why, with the gradual progress of the human species, it can come about, perhaps after many centuries, that no one will any more be praised for accuracy of judgment; for the analytic art (which is still virtually confined to mathematics in its correct and general use) will have become universal and applied to every type of matter through the introduction of a scientific notation (‘philosophical character’) such as I am working on. Once this has been accepted, correct reasoning, given time for thought, will be no more praiseworthy than calculating large numbers without an error. Furthermore, if there is also a trustworthy catalogue of facts (records, observations, experiments) written in the same notation, together with the more important theorems (to reduce the number of steps needed) derived from the notation either alone or with observational data, it will come about that the art of combination will also lose all its glory. Nor, too, will any respect be paid to those who have the opportunity to investigate or discover something by devoting time to thinking about it (since that will be open to everyone), but only to those who are quick at analytic or combinatoric. . . .
[170] Further, what I have said about the distinction between Combinatoric and Analytic will help to differentiate two types of human mind: those that are more combinatorial, and those that are more analytical. Thus, even though Galileo and Descartes excelled in both arts, there was more combinatoric in Galileo, and more Analytic in Descartes. Geometers and lawyers are more analytical, doctors and those concerned with society more combinatorial. There is more certainty in Analytic, more difficulty in Combinatoric.
Mariotte says that the human mind is like a bag: the process of thinking consists in shaking it until something comes out. So there is undoubtedly a chance element in our thought processes. I would add that the human mind is more analogous to a sieve: the process of thinking consists in shaking it until all the subtlest items pass through. Meanwhile, as they are passing through, Reason acts as an inspector snatching out whatever seems useful. It is just as if someone, in order to arrest a thief, made the whole population of a town parade through one particular gate past the watchful eye of the thief’s victim. But to shorten the proceedings a method of exclusion is used, such as that of transition in arithmetic. Thus, if the victim asserts that it was a man, not a woman; or an adult, not a boy or youth, the latter will be granted permission to go on their way.
2. Untitled, undated note (GP.VII)
[200] But to return to the expression of thoughts by means of characters, I thus think that controversies can never be resolved, nor sectarian disputes be silenced, unless we renounce complicated chains of reasoning in favour of simple calculations, and vague terms of uncertain meaning in favour of determinate characters.
In other words, it must be brought about that every fallacy becomes nothing other than a calculating error, and every sophism expressed in this new type of notation becomes in fact nothing other than a grammatical or linguistic error, easily proved to be such by the very laws of this philosophical grammar.
Once this has been achieved, when controversies arise, there will be no more need for a disputation between two philosophers than there would be between two accountants [computistas]. It would be enough for them to pick up their pens and sit at their abacuses, and say to each other (perhaps having summoned a mutual friend): ‘Let us calculate.’
3. Untitled, undated note (Couturat, pp.186–7).
It is very important to realise that the number of primary propositions is infinite, since they are either definitions or axioms. The number of definitions as well as of terms is infinite. The number of axioms is likewise. By axiom I mean an indemonstrable necessary proposition. Necessary — i.e. of which the contrary implies a contradiction. But the only proposition of which the contrary implies a contradiction without one’s being able to demonstrate it, is one of formal identity. The identity is formulated explicitly in the proposition, so it cannot be demonstrated — demonstrated — i.e. made evident by Reason and inferences. Here the identity can be made visible to the eye, so in this case it cannot be demonstrated. The senses make it evident that ‘A is A’ is a proposition of which the opposite, ‘A is not-A,’ formally implies a contradiction. But that which the senses make evident is indemonstrable. So the real, indemonstrable axioms are identical propositions. But their number is infinite. For since the number of terms is infinite, the number of such propositions is also infinite, because there can be as many of them as there are terms. However, this is amazing, and it would seem strange to someone who was told without explanation that the number of infallible primary propositions is infinite. If the principles are infinite, [187] the conclusions from them will be even more so. Such identical propositions are: ‘everything is as much as it is;’ or ‘anything is equal to itself;’ or ‘anything is similar to itself.’
We cannot easily recognise indefinable primary terms for what they are. They are like prime numbers, which we have hitherto been able to identify only by trying to divide them. Similarly, irresoluble terms could be recognised properly only negatively and provisionally. For I know one criterion by which one can recognise resolubility. This is as follows: when we come across a proposition which looks necessary, but has not been demonstrated, it infallibly follows that this proposition contains a definable term (provided that it is necessary). So we must try to give this demonstration, which cannot be done without finding the definition in question. By this method, letting no axiom go without proof (except definitions and identicals), we shall arrive at the Resolution of Terms, and the ultimately simple ideas. You will say that this could go on to infinity, and that new propositions could always be proved, which would oblige us to look for new resolutions. I do not believe so. But if it were the case, it would not matter, since by this method we would not have failed to have perfectly demonstrated all our theorems, and the resolutions which we would have performed would suffice us for an infinity of valid practical inferences. Just as in natural science we should not abandon experimental research because of its potential infinity, since we can already make perfectly good use of the results we have so far obtained.
4. Linguistic Analysis, 11th September 1678 (Couturat, pp.351–2).
Analysis of thoughts is necessary for the discovery and demonstration of truths, because this will correspond to analysis of the characters we use for signifying our thoughts (since a particular thought will correspond to each character). Hence we can make the analysis of thoughts perceptible, and govern it by a sort of mechanical guiding-thread, since the analysis of characters is something perceptible. Analysis of characters consists in substituting for certain characters other characters which are functionally equivalent to the first, but with the one condition that we substitute many for one, and more for fewer (provided these are not equivalent to each other). It is obvious that the thoughts corresponding to the substituted characters will also be equivalent to the meaning of the original character put forward for resolution. But this is made easier by the use of characters than if we set to work upon our thoughts themselves, with no reference to characters. For our intellect needs to be regulated by some sort of mechanical guiding-thread, on account of its weakness. [352] In the case of thoughts which involve things not representable in imagination, this function is performed by the characters themselves.
Further, all disciplines consisting of demonstrations involve nothing other than equivalences or substitutions of thoughts. For they show that in some necessary proposition the predicate can safely be substituted in place of the subject; and in demonstrating, that in place of certain truths (called premises) there can safely be substituted another (called the conclusion). From this it is obvious that the truths themselves will be exhibited on paper in their correct order only through analysis of characters, i.e. continued orderly substitution.
5. Elements of Reason, c.1686 (Couturat, pp.335ff.)
This century has been fortunate to see the invention of an instrument which is an amazingly useful aid to the eye, itself the most valuable of our bodily organs for acquiring knowledge of things. But this instrument [organon] of Reason which we shall now outline exceeds all telescopes and microscopes by as much as Reason itself, which is the instrument of instruments and, so to speak, the eye of the eye, exceeds not only the eye, but every other natural instrument.
It is certainly not difficult to explain why hitherto only the mathematical disciplines have been developed to the point of wonder and envy, not only for their certainty, but also for their abundance of outstanding truths. This cannot be attributed to the intellects of mathematicians, since it is a common observation that when they wander outside their specialisms they are in no way superior to their fellow humans. No, it lies in the nature of the subject, since no hard work or expensive experiments are needed for it to be set before the eyes in such a way that no doubt is left, and there is revealed a certain sequence, or, so to speak, guiding-thread of thought, which gives us confidence about what has already been discovered, and points out an infallible route towards future discoveries.
This is why the perfection of physical science (experience apart) uncontroversially consists in its reduction to geometry, by the discovery of mechanisms (as far as that sort of thing can exist) which depend on the shapes and motions of their parts. But in its turn, geometry itself has up to now been subject to no little confusion, since not all characteristics of figures can be appropriately represented by lines drawn on paper. So it has been reduced to a sort of calculus or numerical computation, so that the very figures of bodies can be expressed by various combinations of numerical characters, and letters of the alphabet standing for indeterminate numbers. [336] This marvellous method is commonly called the ‘specious’ calculus, because of the characters, i.e. species of things. Nothing is more appropriate, easier, or more within the grasp of the human intellect than numbers themselves. The science of number has acquired a higher degree of perfection, and can acquire yet more, through the Combinatory (or general specious) Art, which has given rise to the mathematicians’ ‘analysis’ through its application to numbers. Yet proofs of every analytic truth can always be established by ordinary numbers. So much so, that I have worked out a method of testing every algebraic calculation by abjection of the novenary or the such like, in the manner of the common calculus. This means that every pure mathematical truth can, by means of numbers, be transferred from Reason to visual experiment.
Hitherto, other areas of human reasoning have lacked this advantage of a perpetual experiential test and a perceptible guiding-thread through the labyrinth of thinking, such as can be seen by the eyes and as it were felt by the hands (and in my opinion it is to these factors that mathematics owes its success). For in physics, experiments are difficult, expensive, and fallible; in ethical and social studies they are doubtful and hazardous — or rather both in each case; in metaphysics (the science of incorporeal substances apart from our own) experiments are, in this life, generally not even possible, and are made up for only by grace of divine faith. . . .
[337] So we only have to work out how some sort of instrument can be provided to do for the mind what theodolite and line do for the surveyor, scales for the assayer, number for the mathematician, [338] or telescope for the eye. That is, not only to guide us in judging, but to lead us on to discovery.
It certainly cannot be denied that the ancients achieved much in this area. Even before Plato, there was some not inconsiderable practice of the art of dialectic, as can be gathered even from his dialogues. But as far as is known, it was Aristotle who, standing on the shoulders of his predecessors, was the first to grace logic itself with the form of a mathematical discipline, so as to be amenable to demonstrations. On that account, or because of his example, I admit that the human species owes him a great debt. However, he himself seems to have used too little of such logic outside logic itself, and he certainly did not know how it was possible to apply the same principles to metaphysics, ethics, and all other areas of reasoning intrinsically independent of sensuous imagery. By using some form of combinatory art, one could advance far enough to use substitute characters and letters of the alphabet to bring these subjects under the scope of imagery, like numbers and algebra. Unless I am mistaken, this has so far remained a secret, and is now emerging for the first time.
Further, even the following cannot be denied: if, in reasoning as well as in arguing, people always used the logicians’ forms with inexorable and unflagging rigidity, assuming nothing as true which had not been previously proved by experience of properly marshalled arguments, then they could at least avoid error in reasoning, and, where they could not arrive at truths, at least they would avoid uttering falsehoods, and would even demonstrate many things which are now considered doubtful. However, this rigour in argumentation is more difficult than one might think, especially because of the extremely deceptive ambiguities of the words people use, and the almost insuperable boredom of long-windedness and bickering encountered by anyone who tries to construct a long chain of reasons in the manner of reasoning accepted in the universities. We see that most people have hardly enough patience for thinking about obvious and easy issues — so how much less when prolixity and difficulty are also involved!
The position is made worse by the false belief that, strictly speaking, no form of argument can be approved unless it follows the childish scholastic formulae of the universities, and reeks of their ‘Barbara’ and their ‘Baroco’ [nonsense mnemonics for valid syllogistic forms]. But in my opinion, every argument which is valid in virtue of its form (that is, which will always be truth-preserving) [339] still has the correct form when any other instances are substituted for the original instance. So, not only do mathematicians’ demonstrations have their own structure sufficient for validity, but in all everyday life and practice there are many more accurate demonstrations (depending on the nature of the subject-matter) than run-of-the-mill philosophers think. These philosophers measure everything by three-termed syllogisms, and are not sufficiently aware that long chains of arguments are connected and abbreviated in a certain wonderful smoothness characteristic of natural language. This capacity of human speech has been polished by daily use in civilised languages, and belongs especially to the particles, which carry almost all its logical force. And I would go as far as to say that there are innumerable passages in good authors (especially the orators) which have no lack of conclusive force, despite embracing many things. For transposition of sentences does not change essential form, and it cannot be deceitful for a speaker to clothe the intrinsically dry and bloodless skeleton of an argument with flesh and sinews for the sake of persuasive efficiency. . . .
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