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also available: https://telegra.ph/The-cartesian-disease-04-22
From originals published at:
On the cartesian disease (2013)
More on the cartesian method and it's associated disease (2013)
Cartesian method, scientific method and counting problems (2013)
What is, what we understand and what we communicate about it (2014)
On the cartesian disease
The wording "cartesian disease" means an abuse or misuse of the cartesian method. I shall use in the argumentation the final section, because there is a concentrate of the method, formulated in such a precise, equilibrated and astonishingly actual words.
The abuse of some part of the method consists in the excess of use of one ingredient, in parallel with a lack of use of another ingredient provided by Descartes' method.
In the following are examples of cartesian disease, listed according to the place in the citation which is relevant for the abuse characteristic in the respective example. (Please read, however, the whole text in order to identify the countermeasures which the respective abuses ignore.)
- architectural constructs where many, if not most, of people live. The relevant citation from Descartes is:
"Thus it is observable that the buildings which a single architect has planned and executed, are generally more elegant and commodious than those which several have attempted to improve, by making old walls serve for purposes for which they were not originally built. Thus also, those ancient cities which, from being at first only villages, have become, in course of time, large towns, are usually but ill laid out compared with the regularity constructed towns which a professional architect has freely planned on an open plain; so that although the several buildings of the former may often equal or surpass in beauty those of the latter, yet when one observes their indiscriminate juxtaposition, there a large one and here a small, and the consequent crookedness and irregularity of the streets, one is disposed to allege that chance rather than any human will guided by reason must have led to such an arrangement."
Horror, boredom, social problems appeared from living in such functionally designed, but culturally void places, despite the good will of the creators.
- the cohort of dictators, along with their respective ideologies, specific to the 20th century. The relevant citation from Descartes is:
"In the same way I fancied that those nations which, starting from a semi-barbarous state and advancing to civilization by slow degrees, have had their laws successively determined, and, as it were, forced upon them simply by experience of the hurtfulness of particular crimes and disputes, would by this process come to be possessed of less perfect institutions than those which, from the commencement of their association as communities, have followed the appointments of some wise legislator."
- giving value to uninformed, ignorant common sense, encouraging simple reasoning, hence less energy consuming, more viral, over more sophisticated reasoning. Devaluing knowledge. The relevant citation from Descartes is:
"In the same way I thought that the sciences contained in books (such of them at least as are made up of probable reasonings, without demonstrations), composed as they are of the opinions of many different individuals massed together, are farther removed from truth than the simple inferences which a man of good sense using his natural and unprejudiced judgment draws respecting the matters of his experience."
- the almost eradication of geometrical thinking in mathematical education and research, performed mainly in the 20th century, with great success. The relevant citation from Descartes is:
"Then as to the analysis of the ancients and the algebra of the moderns, besides that they embrace only matters highly abstract, and, to appearance, of no use, the former is so exclusively restricted to the consideration of figures, that it can exercise the understanding only on condition of greatly fatiguing the imagination; and, in the latter, there is so complete a subjection to certain rules and formulas, that there results an art full of confusion and obscurity calculated to embarrass, instead of a science fitted to cultivate the mind."
- running for the ultimate explanation, for the grand theory of everything, ultimately confusing the compression technique which is the cartesian method, designed for being able to fit into our small brains this huge reality at once, with the reality itself. The relevant citation from Descartes is:
"The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt."
- abuse of division, creation of a myriad of problems, which is a tremendously efficient technique for advancing little by little our understanding of something, but it has the disadvantage of giving the impression that said problems are the goal and not just a mean for understanding. Excessive division of research interests. A very good resource for nowadays publishers, as well as for thousands and thousands of researchers specialized into solving problems for the sake of it. The relevant passage from Descartes is:
" The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution."
- using, for compression needs, of an unnecessary and unnatural one dimensional formulation of understanding and then thinking exclusively in such terms, forgetting that this streaming is a technique for easier understanding and not a part of the subject of the study. This abuse is present everywhere in CS and probably is the main barrier in front of a better understanding of how the brain, or more generally how the living world works. Neurons have "tasks", edges "are detected" and so on. The relevant passage from Descartes is:
"... assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence. And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted."
More on the cartesian method and it's associated disease
The cartesian method, as described by the passage by Descartes from the final section, is basically a technique of compression and analysis. From wikipedia:
Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384--322 B.C.), though analysis as a formal concept is a relatively recent development.[1]
Some word is from the Ancient Greek ἀνάλυσις (analusis, "a breaking up", from ana- "up, throughout" and lysis "a loosening").[2]
As a formal concept, the method has variously been ascribed to Alhazen,[3] René Descartes (Discourse on the Method), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name or formally describe).
The four rules of the cartesian methods are:
The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.
And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.
In order to justify my claim that the cartesian method is an analysis (or compression) technique, I shall comment these rules one by one.
1. this rule is made by several parts:
"never to accept anything for true which I did not clearly know to be such"
"to comprise nothing more in my judgement than what was presented to my mind"
"so clearly and distinctly as to exclude all ground of doubt"
The first part looks like a thinking hygiene: be sure about your hypotheses.
The second part has to do with the limitations of our brains capacity to process a complex topic. As such, these limitation have nothing to do with the topic under study. Of course we can't advance our understanding of a subject if we can't wholly grasp it in our minds. However, is important to remember that when we splice it in smaller, more understandable parts, we introduce an element which has nothing to do with the subject of study, but with our capacity of understanding (and our prejudices, indeed, as witnessed by the fact that the same research subject is spliced differently in different epochs or places, according to cultural prejudices and not biological "computing power" reasons).
The third part has entirely to do with our limitations. In order to understand the topic, we have to use techniques which "exclude all ground of doubt". The great importance of doubt as a tool for understanding is one of the most viral parts of the cartesian method. It is one of the main ingredients of the scientific method.
____________
2. two parts here as well:
"to divide each of the difficulties under examination into as many parts as possible"
"and as might be necessary for its adequate solution"
While the first part is clearly an analysis technique, the second part tell that the purpose of understanding is to find a solution for a sequence of problems. Each small part, each difficulty has to be solved. To make a comparison, say that we have a huge cake to eat, so we chip at it with our small mouths, claiming that our goal is to well chew each bite.
This is a compression technique: we divide the cake into bites, then, as we chew each bite, we forget about the others. The bad part is that the cake is not just the sum of the bites.
____________
3. This is the most problematic rule, because here is given total priority to the understanding over the subject of understanding. The analysis and compression technique from the rule 2 is taken to extreme: first is suggested something like an eager evaluation
"to conduct my thoughts in such order that"
"by commencing with objects the simplest and easiest to know, I might ascend [...] to the knowledge of the more complex"
"little and little, and, as it were, step by step"
... then, in order to be sure that the eager evaluation works,
"assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence"
otherwise, who cares about the subject of understanding as long as I can produce an working algorithm? Then, we study the algorithm and we forget what was all this about. This looks like the most harmful part of the cartesian method and the main source of the cartesian disease...
____________
4. ... until we read the last rule:
"in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted"
But if we already renounced at the subject of study and we already (recursively) replaced it with an artificial division, enumeration and analysis technique, this rule is only a proclamation of the superiority of understanding of reality over the reality itself: if the understanding of reality is internally coherent, then it is as good as the reality itself.
____________
Conclusion. The cartesian method is designed as a technique for understanding performed by one mind in isolation, severely handicapped by the bad capacity of communication with other isolated minds. It was a very efficient technique, which is now challenged by two effects of its material outcomes:
- better communication channels provided by the net,
- mechanical, or should I say digital, applications of the method which largely surpass the capacity of understanding of one human mind, as witnessed for example by the first computer aided mathematical proofs, or for another example by the fact that we can numerically model physical phenomena, without understanding rigorously why the method works.
____________
If you read this, then you might be interested to read "Descartes, updated" at The "Putnam Program" blog.
Cartesian method, scientific method and counting problems
In my opinion, the best parts of the cartesian method are:
doubt as a tool for advancing understanding, i.e this part of the rule 1:
"never to accept anything for true which I did not clearly know to be such [...] to comprise [...] so clearly and distinctly as to exclude all ground of doubt",
and this part of rule 3, taken out of context, seen as a belief, a state of mind of the researcher:
"by commencing with objects the simplest and easiest to know, I might ascend [...] to the knowledge of the more complex"
This is, in a nutshell, that part of the scientific method which apply even to mathematicians (who don't do experiments). If there is any need to tell, I strongly believe in the scientific method, although I recently understood that I have problems with that parts of the cartesian method which I now think will become obsolete, due to better and faster communication among scientists and due to the confrontation with big data (which will require techniques adapted to the recognition of the fact that reality might be complex enough so that a model of it does not fit wholly into one human brain).
The following are just doubts and questions which might be naive or based on ignorance, but nevertheless I write them here, in the hope of informed comments.
Remember again I am a mathematician, so maybe I disappoint some readers (I hope not my readers) by saying that I have no problem with Cantor diagonal argument for the proof of the fact that the set of reals is uncountable. What makes me feel less comfortable is the impression, which might be false, that famous theorems in logic, like Godel, or Turing, show in fact that there are limits to the enumeration part of the cartesian method. Am I right? To make a comparison, suppose I'am a physicist with infinite powers and I say: I made the experiment of counting all them Turing machines and I get results which are contradictory with older experiments. I deduce then, by the scientific method, that the counting procedure itself, any I might think about, is flawed, because all the other parts have been checked independently. It means I am not allowed to use the part of the cartesian method which comprises enumerations if I want to understand the reality. Reality is like this, point. If I count then I am led astray, like, as another comparison, if I suppose that a quantum object has a arbitrarily well localized position and moment, then I am led astray. (Not that I think there is any connection between logic theorems based on Cantor diagonal argument and quantum mechanics.)
So, what is left of logic if counting arguments are eliminated? Anybody knows? I hope so.
What is, what we understand and what we communicate about it
One of the major effects of the existence of the Net is that it challenges the cartesian method and shatters the foundations of science. As the Net is still young and the traditions are old, comparatively, the effect is not yet clearly identified. But it is there already, and even if not named, it is like a secret ailment which manifests here and there, stronger and stronger.
That which provokes the disease is a lack of balance in the use of the various ingredients of the cartesian method. The disbalance is provoked by the effort to fit what is, what we understand and what we communicate about it into the mould of the era when the method has been invented. It just does not fit, therefore it overflows in unexpected ways.
What is. Better is to say what is more, compared with the time when the cartesian metod was invented. There is a new virtual world in the making. Huge quantities of structured data, alternative worlds evolved from seeds created by programmers, a whole new world of the Internet of Things in the making and, further away but really close though, an unification of the real world (defined as the one where Descartes lived and died) with these new, emerging ones.
The territory, suddenly, got much bigger.
What we understand. Better to say what we understand more than before. A huge body of scientific facts and discoveries which don't quite fit ones with the others. Quantum mechanics with general Relativity has become an example for old boys and girls. We have models of the parts but we don't understand C. Elegans with its 302 neurons. There is though a building understanding of the fact that we do understand much more, but the tools offer, each, only a limited point of view. There are looming suspicions that data alone (what is) has more to tell us than data filtered by a theory. We do understand, or starting to understand that semantics is only a tool itself, a map maker, not the territory of what is.
The many maps we have don't fit, we understand.
What we communicate. Better is to say that we communicate today in unparalleled ways than before. But what we communicate is a very small, rigidly formatted part of what we understand. It is hard to communicate science, and the channel constraints are damaging the message.
We have so much to communicate and the semantic maps don't serve well this purpose.
Going back at the time of the cartesian method, we see that it has been made as a tool for isolated minds and very limited data inflow. More than this, the cartesian method is a collection of prescriptions about how to better understand and how to better communicate the understanding, which makes the rational choice for those times, but an irrational one for our times:
- it privileges what we understand over what is,
- what we communicate over what we understand.
Then it optimizes what we communicate for the situation of one human mind which lays down the output in a book.
The book is then supposed to be distributed and multiplied by means which are not of interest for the author, and then to hit other minds in an unidirectional way.
Descartes writes a book, then somebody else writes another book where he challenges or supports the ideas of a Dead Descartes Book, then yet another one writes a new book which contains references to the Dead Somebody Else Book. That's the way of the science and Descartes proposed a wonderful path which ensures that the various Books are well written and contain Text as a sort of a program which can be easily debugged.
Descartes rules apply in an indiscriminate way to what is, what we understand and what we communicate about it.
Reference: The cartesian method
From Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences, by René Descartes, beginning of Part II.
"I was then in Germany, attracted thither by the wars in that country, which have not yet been brought to a termination; and as I was returning to the army from the coronation of the emperor, the setting in of winter arrested me in a locality where, as I found no society to interest me, and was besides fortunately undisturbed by any cares or passions, I remained the whole day in seclusion, with full opportunity to occupy my attention with my own thoughts. Of these one of the very first that occurred to me was, that there is seldom so much perfection in works composed of many separate parts, upon which different hands had been employed, as in those completed by a single master. Thus it is observable that the buildings which a single architect has planned and executed, are generally more elegant and commodious than those which several have attempted to improve, by making old walls serve for purposes for which they were not originally built. Thus also, those ancient cities which, from being at first only villages, have become, in course of time, large towns, are usually but ill laid out compared with the regularity constructed towns which a professional architect has freely planned on an open plain; so that although the several buildings of the former may often equal or surpass in beauty those of the latter, yet when one observes their indiscriminate juxtaposition, there a large one and here a small, and the consequent crookedness and irregularity of the streets, one is disposed to allege that chance rather than any human will guided by reason must have led to such an arrangement. And if we consider that nevertheless there have been at all times certain officers whose duty it was to see that private buildings contributed to public ornament, the difficulty of reaching high perfection with but the materials of others to operate on, will be readily acknowledged. In the same way I fancied that those nations which, starting from a semi-barbarous state and advancing to civilization by slow degrees, have had their laws successively determined, and, as it were, forced upon them simply by experience of the hurtfulness of particular crimes and disputes, would by this process come to be possessed of less perfect institutions than those which, from the commencement of their association as communities, have followed the appointments of some wise legislator. It is thus quite certain that the constitution of the true religion, the ordinances of which are derived from God, must be incomparably superior to that of every other. And, to speak of human affairs, I believe that the pre-eminence of Sparta was due not to the goodness of each of its laws in particular, for many of these were very strange, and even opposed to good morals, but to the circumstance that, originated by a single individual, they all tended to a single end. In the same way I thought that the sciences contained in books (such of them at least as are made up of probable reasonings, without demonstrations), composed as they are of the opinions of many different individuals massed together, are farther removed from truth than the simple inferences which a man of good sense using his natural and unprejudiced judgment draws respecting the matters of his experience. And because we have all to pass through a state of infancy to manhood, and have been of necessity, for a length of time, governed by our desires and preceptors (whose dictates were frequently conflicting, while neither perhaps always counseled us for the best), I farther concluded that it is almost impossible that our judgments can be so correct or solid as they would have been, had our reason been mature from the moment of our birth, and had we always been guided by it alone.
It is true, however, that it is not customary to pull down all the houses of a town with the single design of rebuilding them differently, and thereby rendering the streets more handsome; but it often happens that a private individual takes down his own with the view of erecting it anew, and that people are even sometimes constrained to this when their houses are in danger of falling from age, or when the foundations are insecure. With this before me by way of example, I was persuaded that it would indeed be preposterous for a private individual to think of reforming a state by fundamentally changing it throughout, and overturning it in order to set it up amended; and the same I thought was true of any similar project for reforming the body of the sciences, or the order of teaching them established in the schools: but as for the opinions which up to that time I had embraced, I thought that I could not do better than resolve at once to sweep them wholly away, that I might afterwards be in a position to admit either others more correct, or even perhaps the same when they had undergone the scrutiny of reason. I firmly believed that in this way I should much better succeed in the conduct of my life, than if I built only upon old foundations, and leaned upon principles which, in my youth, I had taken upon trust. For although I recognized various difficulties in this undertaking, these were not, however, without remedy, nor once to be compared with such as attend the slightest reformation in public affairs. Large bodies, if once overthrown, are with great difficulty set up again, or even kept erect when once seriously shaken, and the fall of such is always disastrous. Then if there are any imperfections in the constitutions of states (and that many such exist the diversity of constitutions is alone sufficient to assure us), custom has without doubt materially smoothed their inconveniences, and has even managed to steer altogether clear of, or insensibly corrected a number which sagacity could not have provided against with equal effect; and, in fine, the defects are almost always more tolerable than the change necessary for their removal; in the same manner that highways which wind among mountains, by being much frequented, become gradually so smooth and commodious, that it is much better to follow them than to seek a straighter path by climbing over the tops of rocks and descending to the bottoms of precipices.
Hence it is that I cannot in any degree approve of those restless and busy meddlers who, called neither by birth nor fortune to take part in the management of public affairs, are yet always projecting reforms; and if I thought that this tract contained aught which might justify the suspicion that I was a victim of such folly, I would by no means permit its publication. I have never contemplated anything higher than the reformation of my own opinions, and basing them on a foundation wholly my own. And although my own satisfaction with my work has led me to present here a draft of it, I do not by any means therefore recommend to every one else to make a similar attempt. Those whom God has endowed with a larger measure of genius will entertain, perhaps, designs still more exalted; but for the many I am much afraid lest even the present undertaking be more than they can safely venture to imitate. The single design to strip one's self of all past beliefs is one that ought not to be taken by every one. The majority of men is composed of two classes, for neither of which would this be at all a befitting resolution: in the first place, of those who with more than a due confidence in their own powers, are precipitate in their judgments and want the patience requisite for orderly and circumspect thinking; whence it happens, that if men of this class once take the liberty to doubt of their accustomed opinions, and quit the beaten highway, they will never be able to thread the byway that would lead them by a shorter course, and will lose themselves and continue to wander for life; in the second place, of those who, possessed of sufficient sense or modesty to determine that there are others who excel them in the power of discriminating between truth and error, and by whom they may be instructed, ought rather to content themselves with the opinions of such than trust for more correct to their own reason.
For my own part, I should doubtless have belonged to the latter class, had I received instruction from but one master, or had I never known the diversities of opinion that from time immemorial have prevailed among men of the greatest learning. But I had become aware, even so early as during my college life, that no opinion, however absurd and incredible, can be imagined, which has not been maintained by some on of the philosophers; and afterwards in the course of my travels I remarked that all those whose opinions are decidedly repugnant to ours are not in that account barbarians and savages, but on the contrary that many of these nations make an equally good, if not better, use of their reason than we do. I took into account also the very different character which a person brought up from infancy in France or Germany exhibits, from that which, with the same mind originally, this individual would have possessed had he lived always among the Chinese or with savages, and the circumstance that in dress itself the fashion which pleased us ten years ago, and which may again, perhaps, be received into favor before ten years have gone, appears to us at this moment extravagant and ridiculous. I was thus led to infer that the ground of our opinions is far more custom and example than any certain knowledge. And, finally, although such be the ground of our opinions, I remarked that a plurality of suffrages is no guarantee of truth where it is at all of difficult discovery, as in such cases it is much more likely that it will be found by one than by many. I could, however, select from the crowd no one whose opinions seemed worthy of preference, and thus I found myself constrained, as it were, to use my own reason in the conduct of my life.
But like one walking alone and in the dark, I resolved to proceed so slowly and with such circumspection, that if I did not advance far, I would at least guard against falling. I did not even choose to dismiss summarily any of the opinions that had crept into my belief without having been introduced by reason, but first of all took sufficient time carefully to satisfy myself of the general nature of the task I was setting myself, and ascertain the true method by which to arrive at the knowledge of whatever lay within the compass of my powers.
Among the branches of philosophy, I had, at an earlier period, given some attention to logic, and among those of the mathematics to geometrical analysis and algebra,--three arts or sciences which ought, as I conceived, to contribute something to my design. But, on examination, I found that, as for logic, its syllogisms and the majority of its other precepts are of avail--rather in the communication of what we already know, or even as the art of Lully, in speaking without judgment of things of which we are ignorant, than in the investigation of the unknown; and although this science contains indeed a number of correct and very excellent precepts, there are, nevertheless, so many others, and these either injurious or superfluous, mingled with the former, that it is almost quite as difficult to effect a severance of the true from the false as it is to extract a Diana or a Minerva from a rough block of marble. Then as to the analysis of the ancients and the algebra of the moderns, besides that they embrace only matters highly abstract, and, to appearance, of no use, the former is so exclusively restricted to the consideration of figures, that it can exercise the understanding only on condition of greatly fatiguing the imagination; and, in the latter, there is so complete a subjection to certain rules and formulas, that there results an art full of confusion and obscurity calculated to embarrass, instead of a science fitted to cultivate the mind. By these considerations I was induced to seek some other method which would comprise the advantages of the three and be exempt from their defects. And as a multitude of laws often only hampers justice, so that a state is best governed when, with few laws, these are rigidly administered; in like manner, instead of the great number of precepts of which logic is composed, I believed that the four following would prove perfectly sufficient for me, provided I took the firm and unwavering resolution never in a single instance to fail in observing them.
The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.
And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.
The long chains of simple and easy reasonings by means of which geometers are accustomed to reach the conclusions of their most difficult demonstrations, had led me to imagine that all things, to the knowledge of which man is competent, are mutually connected in the same way, and that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it, provided only we abstain from accepting the false for the true, and always preserve in our thoughts the order necessary for the deduction of one truth from another. And I had little difficulty in determining the objects with which it was necessary to commence, for I was already persuaded that it must be with the simplest and easiest to know, and, considering that of all those who have hitherto sought truth in the sciences, the mathematicians alone have been able to find any demonstrations, that is, any certain and evident reasons, I did not doubt but that such must have been the rule of their investigations. I resolved to commence, therefore, with the examination of the simplest objects, not anticipating, however, from this any other advantage than that to be found in accustoming my mind to the love and nourishment of truth, and to a distaste for all such reasonings as were unsound. But I had no intention on that account of attempting to master all the particular sciences commonly denominated mathematics: but observing that, however different their objects, they all agree in considering only the various relations or proportions subsisting among those objects, I thought it best for my purpose to consider these proportions in the most general form possible, without referring them to any objects in particular, except such as would most facilitate the knowledge of them, and without by any means restricting them to these, that afterwards I might thus be the better able to apply them to every other class of objects to which they are legitimately applicable. Perceiving further, that in order to understand these relations I should sometimes have to consider them one by one and sometimes only to bear them in mind, or embrace them in the aggregate, I thought that, in order the better to consider them individually, I should view them as subsisting between straight lines, than which I could find no objects more simple, or capable of being more distinctly represented to my imagination and senses; and on the other hand, that in order to retain them in the memory or embrace an aggregate of many, I should express them by certain characters the briefest possible. In this way I believed that I could borrow all that was best both in geometrical analysis and in algebra, and correct all the defects of the one by help of the other."