Date: 1st July 2021
Presenter: Sebastjan Vörös
Chapters: Ch. I “The Origins of Modern Science” & Ch. II “Mathematics as an Element in the History of Thought”
Keywords: mathematics, evil, relations, concrescence, God, potentiality, actuality, creativity
Discussion
Summary
33:59 (Matthias) – Milton’s Paradise Lost in connection to the first chapter; the character of Satan (the most flashed out character of the entire epic); we should not aspire to the same level of knowledge which only God can possess (there is a limit to what we can know); in demystifying the notion of forbidden knowledge, Satan says the following:
A mind not to be chang’d by Place or Time.
The mind is its own place, and in it self
Can make a Heav’n of Hell, a Hell of Heav’n.
What matter where, if I be still the same,
And what I should be, all but less then he
Whom Thunder hath made greater?
[Paradise Lost, Book I, lines 221-270]
Here Satan is comparing himself to God; for him the only difference is that of magnitude of power; there exists a certain tension in the western tradition between the hunger for knowledge and humbleness with regards to what we cannot (or should not) know.
48:20 (Miha) – Whitehead’s understanding of the devil when he was still a student at Cambridge and a member of the Apostles (a debating/discussing club)
50:17 (Sebastjan) – The reintroduction of capacities of deductive reasoning without the loss of rigor or the necessary capitulation of reason (or philosophy as such).
51:17 (Miha) – Imaginative rationalisation as used by Whitehead in Process and Reality; Giordano Bruno
52:15 (Primož) – Whitehead is more interested in the how of science not the why.
53:38 (Gabor) – “A chain of facts is like a barrier reef. On one side there is wreckage, and beyond it harbourage and safety. The categories governing the determination of things are the reasons why there should be evil; and are also the reasons why, in the advance of the world, particular evil facts are finally transcended.” [PR 223]
55:28 (Chris) – Prohibition against curiosity which had its origin in the ancient Greek philosophy (the tale of Thales who fell down a well when gazing up at the heavens); later picked up by Augustine; Whitehead’s take on the reformation and his almost pedagogical use of historical examples; Whitehead’s understanding of the relation between science and philosophy
1:04:34 (Sebastjan) – Thought style
1:06:24 (Gabor) – Ch. IX of Science and the Modern World
1:10:29 (Gabor) – Whitehead’s conception of evil in Process and Reality: “The revolts of destructive evil, purely self-regarding, are dismissed into their triviality of merely individual facts; and yet the good they did achieve in individual joy, in individual sorrow, in the introduction of needed contrast, is yet saved by its relation to the completed whole. The image—and it is but an image—the image under which this operative growth of God’s nature is best conceived, is that of a tender care that nothing be lost.” [PR 346]
1:10:50 (Gabor) – Consequent nature of God: “The consequent nature of God is his judgment on the world. He saves the world as it passes into the immediacy of his own life. It is the judgment of a tenderness which loses nothing that can be saved. It is also the judgment of a wisdom which uses what in the temporal world is mere wreckage.” [PR 346]
1:20:28 (Gabor) – On Ch. II.: Why did Whitehead choose to talk about mathematics?
1:23:03 (Hridija) – Whitehead, Einstein’s relativity theory and Bergson
1:28:02 (Sebastjan) – The slow entering of organic metaphors into the realms of science (as well as philosophy); “That a meeting-point between biology and physical science may at some time be found there is no reason for doubting. But we may confidently predict that if that meeting point is found, and one of the two sciences is swallowed up, that one will not be biology.” [John Scott Haldane]; the revitalisation of the concept of life.
1:32:02 (Miha) – Whiteheads notion of “relations” as presented in chapters I. and II.
1:36:48 (Timotej) – What is the relation between Whitehead’s mathematical (Platonistic) position on the one hand and his holistic (referring to the circular causality between the whole and the parts) position on the other?
1:39:37 (Gabor) – On eternal objects; for Whitehead they represent potentialities (seeds for actuality)
1:42:00 (Sebastjan) – Potentiality understood as a horizon; Plessner’s understanding of a “surplus of negativity” which differentiates humans from animals; the ability of grasping the negative; another connection might be that of potentiality and virtuality.
1:44:28 (Matthias) – We can understand the particular occurrences through high abstraction; referring here to Ch. II. p. 27-28 (Free Press 1967): “[F]rom a select set of those general conditions, exemplified in any one and the same occasion, a pattern involving an infinite variety of other such conditions, also exemplified in the same occasion, can be developed by the pure exercise of abstract logic. Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.”
1:47:25 (Hridija) – Whitehead “cannot part from his own shadow” [cf. 20, Free Press Edition] of being a mathematician; the importance of the concept of concrescence as a bridge between the abstract and the concrete; “[I]n being a mathematician he [Whitehead] is more than a mathematician” [cf. 20]
1:49:09 (Gabor) – The concept of concrescence and how it relates to potentiality and actuality; concrescence is the way by which the many of the universe (potentialities) become something new (actuality)
1:52:02 (Hridija) – Subject and superject
1:53:42 (Matthias) – What is the process through which certain things get actualised and others disclosed? Is it at all similar to Buddhist conceptions?
1:54:55 (Gabor) – “[S]ubjectivity is the very process of self-determination by selecting from potentialities, becoming a superject in the end (not subject)”; for Whitehead there is no subject in the sense of a pre-existing entity; subjectivity is a process (of experiencing); by use of “subjective aim” the subject “pulls itself from the mud by its own hair”; the notion of satisfaction; temporality; creativity.
2:01:25 (Sebastjan) – Realism or idealism?
2:04:42 (Izak) – Whitehead’s outdated notions of mathematics, especially numbers (for example the number three is a set of all the sets which contain three elements); Does Whitehead at all tackle the question of how philosophy could influence the development of mathematics? Whitehead as a kind of “constructive structuralist”; Whitehead’s use of “practical abstractness,” not just abstractness as is.
2:12:46 (Gabor) – Whitehead turned to mathematics several times in his writings: in 1911 he wrote An Introduction to Mathematics and in 1941 he wrote Mathematics and the Good, and also in parts of Concept of Nature and Process and Reality
Plant of the week:
Narrow-leaved flax (Linum teniufolium; sl. drobnolistni lan) grows on dry limestone meadows, screes, and in rocky steppes. Its range spans from Spain in the west to Iran in the east, and from the Mediterranean regions of Europe in the south to Germany and Poland in the north. The specimens pictured here were found east of Kuželjska stena above the Kolpa valley (river Kolpa is a natural border between southeast Slovenia and northwest Croatia).
Flaxseed and linen are obtained from the common flax or linseed (Linum usitatissimum; sl. navadni ali evrazijski lan). Interestingly, this species is known only as a cultivated plant, and appears to have been domesticated from the pale flax (Linum bienne, sl. dvoletni lan) in the Fertile Crescent cca. 9000 years ago. It is quite similar in appearance to narrow-leaved flax, but its large white flowers are blue-veined instead of pink-veined. It used to be cultivated in Ancient Egypt; most of its production in the Middle Ages was in Flanders; it was also introduced to North America by the colonists. Due to cheap cotton and rising farm wages in the early 20th century, its production in America and Europe declined, and since then northern Russia provided more than 90 % of the crops. Today, Kazakhstan is its largest producer, cultivating almost one third of the world’s linseed.
Abridgment of Chs. I & II
By: Miha Flere
Chapter I: The Origins of Modern Science, p. 1-24
(p. 1) According to Whitehead the progress of civilisation is not wholly a uniform drift toward better things. It may perhaps wear this aspect if we map it on a scale which is large enough. But such broad views obscure the details on which rest our whole understanding of the process.
The sixteenth century of our era saw the disruption of Western Christianity and the rise of modern science. It was an age of ferment. In science, Copernicus and Vesalius may be chosen as representative figures: they typify the new cosmology and the scientific emphasis on direct observation.
A rather special emphasis is given to Giordano Bruno, who, for Whitehead, was the martyr; though the cause for which he suffered was not that of science, but that of free imaginative speculation.
(p. 2) With regards to religion (a) the Reformation, for all its importance, may be considered as a domestic affair of the European races. We cannot look upon it as introducing a new principle into human life. It was a great transformation of religion—it was a popular uprising, and for a century and a half drenched Europe in blood—but it was not the coming of religion. In fact, reformers maintained that they were only restoring what had been forgotten.
In contrast to religion Whitehead mentions the rise of modern science (b). The beginnings of the scientific movement were confined to a minority among the intellectual elite. The worst that happened to men of science was that Galileo suffered an honourable detention and a mild reproof, before dying peacefully in his bed. This for Whitehead is characteristic of how the rise of modern science is in reality just the quiet commencement of the most intimate change in outlook which the human race had yet encountered—yet (p. 3) this slight change in tone makes all the difference.
This new tinge to modern minds is a vehement and passionate interest in the relation of (i) general principles to (ii) irreducible and stubborn facts. This balance of minds (i, ii) has now become part of the tradition which infects cultivated thought—this will be called the first contrast of science. It is the salt which keeps life sweet. The second contrast, as Whitehead points out, is the universality of science.
(p. 4) Whitehead gives further emphasis to the fact that in this book he will not be discussing the details of scientific discovery. On the contrary the theme will be the energising of a state of mind in the modern world, its broad generalisations, and its impact upon other spiritual forces.
The history of thought should be read in two ways, both forwards (f) and backwards (b). A “climate of opinion” (Joseph Glanville) therefore requires for its understanding the consideration of its antecedents (b) and its issues (f). In this lecture Whitehead shall consider some of the antecedents of our modern approach to the investigation of nature.
Since the time of Hume, the fashionable scientific philosophy has been as to deny the rationality of science. This conclusion, for Whitehead, lies in Hume’s take on the concept of causation. (p. 5) (i) If the cause in itself discloses no information as to the effect, so that the first invention of it must be entirely arbitrary, it follows at once that science is impossible, except in the sense of establishing entirely arbitrary connections. (ii) But scientific faith has risen to the occasion, and has tacitly removed the philosophic mountain. In view of this strange contradiction (i, ii) in scientific thought, it is of the first importance to consider the antecedents of a faith which is impervious to the demand for a consistent rationality. We have therefore to trace the rise of the instinctive faith that there is an Order of Nature which can be traced in every detailed occurrence.
Of course we all share in this faith, says Whitehead, and we therefore believe that the reason for the faith is our apprehension of its truth. But the formation of a general idea—such as the idea of the Order of Nature—and the grasp of its importance, and the observation of its exemplification in a variety of occasions, are by no means necessary consequences of the truth of the idea in question. Familiar things happen, and mankind does not bother about them. It requires a very unusual mind to undertake the analysis of the obvious. Accordingly, Whitehead wishes to consider the stages in which this analysis became explicit.
(p. 6) Whitehead turns his attention to two very important aspects that man has observed in nature. (i) In broad outline certain general states of nature recur, and our natures have adapted themselves to such repetition. But there is a complementary fact which is equally true and equally obvious: (ii) nothing ever really recurs in exact detail. No two days are identical, no two winters. What has gone, has gone for ever. Men expected the sun to rise, but the wind bloweth where it listeth (John 3:8).
Certainly from the classical Greek civilisation onwards there have been men, and indeed groups of men, who have placed themselves beyond this acceptance of an ultimate irrationality. Such men have endeavoured to explain all phenomenon as the outcome of an order of things which extends to every detail. But until the close of the Middle Ages the general educated public did not feel that intimate conviction. (p. 7) That being sad, why did the pace suddenly quicken in the sixteenth and seventeenth centuries? At the close of the Middle Ages a new mentality discloses itself—invention stimulated thought, thought quickened physical speculation, Greek manuscripts disclosed what the ancients had discovered.
There have been great civilisations in which the peculiar balance of mind required for science has only fitfully appeared and has produced the feeblest result. Whitehead gives the example of China which is to be admired for the heights of its civilisation. There is no reason to doubt the intrinsic capacity of individual Chinamen for the pursuit of science. And yet Chinese science is practically negligible—the same goes for India.
Even as it was, the Greeks, though they founded the movement [of modern scientific thought], did not sustain it with the concentrated interest which modern Europe has shown. (p. 9) In spite of all their advances in strict reasoning, clear and bold ideas, this was not science as we understand it. The patience of minute observation was not nearly so prominent. Their genius was not so apt for the state of imaginative muddle suspense which precedes successful inductive generalization.
That being said, every philosophy, even Greek, is tinged with the colouring of some secret imaginative background, which never emerges explicitly into its trains of reasoning. For the Greeks, Whitehead points out, it was their overwhelmingly dramatic view of nature—(p. 10) nature was a drama in which each thing played its part.
The effect of such an imaginative setting for nature was to damp down the historical spirit. For it was the end which seemed illuminating, so why bother with the beginning? As Whitehead points out, it is not a view that Aristotle would have necessarily subscribed to. It was a view which subsequent Greek thought extracted from Aristotle and passed on to the Middle Ages.
The Reformation and the scientific movement were two aspects of the historical revolt which was the dominant intellectual movement of the later Renaissance.
It is a great mistake to conceive this historical revolt as an appeal to reason. On the contrary, it was through and through an anti-intellectualist movement. It was the return to the contemplation of brute fact; and it was based on a recoil from the inflexible rationality of the medieval thought.
(p. 11-12) In support to this, Whitehead points to two sources: (i) to the forth book of Father Paul Sarpi’s History of the Council of Trent and (ii) Richard Hooker’s Laws of Ecclesiastical Polity. Both of them testify to the anti-rationalist trend of thought at that epoch, and in this respect contrast their own age with the epoch of scholasticism.
It will take centuries before stubborn facts are reducible by reason, and meanwhile the pendulum swings slowly and heavily to the extreme of the historical method.
Although, Whitehead says, one outcome of this reaction was the birth of modern science, yet we must remember that science thereby inherited the bias of thought to which it owes its origin—that is the aforementioned Greek imaginative background of the dramatic view of nature.
Their vision of fate, remorseless and indifferent, urging a tragic incident to its inevitable issue, is the vision possessed by science. Fate in Greek Tragedy becomes the order of nature in modern science.
(p. 13) On this point, Whitehead wants to remind us, that the essence of dramatic tragedy is not unhappiness. It resides in the solemnity of the remorseless working of things. This remorseless inevitableness is what pervades scientific thought. The laws of physics are the decrees of fate.
This can be furthered by the idea of moral order, whose spectacle was impressed upon the imagination of classical civilisation.
(p. 14) Later on the concept of moral order and of the order of nature had enshrined itself in the Stoic philosophy. In support to this, Whitehead mentions a passage from Lecky’s History of European Morals, which goes as follows: “Seneca maintains that the Divinity has determined all things by an inexorable law of destiny, which He has decreed, but which He Himself obeys.” The most effective way in which the Stoics influenced the mentality of the Middle Ages was by the diffused sense of order which arose from Roman law. Again, Whitehead quotes Lecky: “The Roman legislation was in a twofold manner the child of philosophy. It was in the (i) first place formed upon the philosophical model, for, instead of being a mere empirical system adjusted to the existing requirements of society, it laid down abstract principles of right to which it endeavoured to conform; (ii) and, in the next place, these principles were borrowed directly from Stoicism.”
In the Middle Ages it became the conception of a definite articulated system which defines the legality of the detailed structure of social organism, and of the detailed way in which it should function. It was not a question of admirable maxims, but of definite procedure to put things right and to keep them there.
(p. 15) But for science more is wanted than a general sense of the order of things.
The greatest contribution of the medievalism to the formation of the scientific movement was the inexpungable belief that every detailed occurrence can be correlated with its antecedents in a perfectly definite manner, exemplifying general principles. It is this instinctive conviction, vividly poised before the imagination, which is the motive power of research: that there is a secret, a secret which can be unveiled.
(p. 16-18) In Asia, the conceptions of God were of a being who was either too arbitrary or too impersonal for such ideas to have much effect on instinctive habits of mind. There was not the same confidence as in the intelligible rationality of a personal being—the rationality of God, conceived as with the personal energy of Jehovah and with the rationality of a Greek philosopher. This being said, Whitehead is not arguing that the European trust in the scrutability of nature was logically justified even by its own theology.
Whitehead’s explanation is that the faith in the possibility of science, generated antecedently to the development of modern scientific theory, is an unconscious derivative from medieval theology.
But science is not merely the outcome of instinctive faith. It also requires an active interest in the simple occurrences of life for their own sake—this qualification “for their own sake”, Whitehead deems especially important.
(1) The first phase of the Middle Ages was an age of symbolism. Art, in particular, has a haunting charm beyond compare: its own intrinsic quality is enhanced by the fact that its message, which stretched beyond art’s own self-justification of aesthetic achievement, was the symbolism of things behind nature itself. In this symbolic phase, medieval art energised in nature as its medium, but pointed to another world.
(2) The codification of Roman law established the ideal of legality. Law is both an engine for government, and a condition restraining government. This established that an authority should be at once lawful, and law-enforcing, and should in itself exhibit a rationally adjusted system of organisation.
(3) In the non-political spheres of art and learning Constantinople exhibited a standard of realised achievement which, (a) partly by the impulse to direct imitation, and (b) partly by the indirect inspiration arising from the mere knowledge that such things existed, acted as a perpetual spur to Western culture.
(p. 19) Gregory the Great and (especially) St Benedict contributed elements to the reconstruction of Europe which secured that this reconstruction, when it arrived, should include a more effective scientific mentality than that of the ancient world. The Greeks were over-theoretical. For them science was an offshoot of philosophy. They were practical man, with an eye for the importance of ordinary things; and they combined this practical temperament with their religious and cultural activities. The alliance of science with technology, by which learning is kept in contact with irreducible and stubborn facts, owes much to the practical bent of the early Benedictines—the monasteries were the homes of practical agriculturalists, as well as of saints and of artists and men of learning.
The rise of Naturalism in the later Middle Ages was the entry into the European mind of the final ingredient necessary for the rise of science. It was the rise of interest in natural objects and natural occurrences, for their own sakes.
(p. 20) The growth of wealth and leisure; the expansion of universities, the invention of printing; the taking of Constantinople; Copernicus; Vasco da Gama; Columbus; the telescope. The soil, the climate, the seeds were there, and the forest grew. Science has never shaken off the impress of its origin in the historical revolt of the later Renaissance. It has remained predominantly an anti-rationalistic movement, based upon a naïve faith.
What reasoning it has wanted, has been borrowed from mathematics which is a surviving relic of Greek rationalism. Science repudiates philosophy. In other words, it has never cared to justify its faith or to explain its meanings, and has remained blandly indifferent to its refutation by Hume.
Of course the historical revolt was fully justified. It was wanted. It was more than wanted: it was absolute necessity for healthy progress—a very sensible reaction; but not a protest on behalf of reason.
(p. 21) The truth is that science started its modern career by taking over ideas derived from the weakest side of the philosophies of Aristotle’s successors. In some respects it was a happy choice. It enabled the knowledge of the seventeenth century to be formularised so far as physics and chemistry were concerned, with a completeness which has lasted to the present time. But the progress of biology and psychology has probably been checked by the uncritical assumption of half-truths.
(p. 22) According to Whitehead, there persists, throughout the whole period the fixed scientific cosmology which presupposes the ultimate fact of an irreducible brute matter, or material, spread throughout space in a flux of configurations. In itself such material is senseless, valueless, purposeless. It just does what it does do, following a fixed routine imposed by external relations which do not spring from the nature of being. It is this assumption that I call “scientific materialism”.
It is not wrong, if properly construed. If we confine ourselves to certain types of facts, abstracted from the complete circumstances in which they occur, the materialistic assumption expresses these facts to perfection. But when we pass beyond the abstraction, either by more subtle employment of our senses, or by the request for meanings and for coherence of thoughts, the scheme breaks down at once.
(p. 23-24) The historical revolt has thus been exaggerated into the exclusion of philosophy from its proper role of harmonising the various abstractions of methodological thought. Thought is abstract; and the intolerant use of abstractions is the major vice of the intellect. This vice is not wholly corrected by the recurrence to concrete experience.
There are two methods for the purification of ideas. (1) One of them is dispassionate observation by means of the bodily senses. But observation is selection. Accordingly, it is difficult to transcend a scheme of abstraction. (2) The other method is by comparing the various schemes of abstraction which are well founded in our various types of experience.
The faith in the order of nature which has made possible the growth of science is a particular example of a deeper faith. This faith cannot be justified by any inductive generalisation. It springs from direct inspection of the nature of things as disclosed in our own immediate present experience. There is no parting from your own shadow. To experience this faith is to know that in being ourselves we are more than ourselves: to know that our experience, dim and fragmentary as it is, yet sounds the utmost depths of reality: to know that detached details merely in order to be themselves demand that they find themselves in a system of things: to know that this system includes the harmony of logical rationality, and the harmony of aesthetic achievement: to know that, while the harmony of logic lies upon the universe as an iron necessity, the aesthetic harmony stands before it as a living ideal moulding the general flux in its broken progress towards finer, subtler issues.
(Edgar Degas, Field of flax, 1892; source: impressionist-art)
Chapter II: Mathematics as an Element in the History of Thought, p. 25-48
(p. 25-27) The originality of mathematics consists in the fact that in mathematical science connections between things are exhibited which, apart from the agency of human reason, are extremely unobvious. Thus the ideas, now in the minds of contemporary mathematicians, lie very remote from any notions which can be immediately derived by perception through the senses; unless indeed it be perception stimulated and guided by antecedent mathematical knowledge.
Omitting mathematics from a history of thought would be like cutting out the part of Ophelia in Hamlet. For Ophelia is quite essential to the play (not as much as the character of Hamlet by which the play is named after), she is very charming, —and a little mad. Let us grant that the pursuit of mathematics is a divine madness of the human spirit, a refuge from the goading urgency of contingent happenings.
The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. All you assert is, that reason insists on the admission that, if any entities whatever have any relations which satisfy such-and-such purely abstract conditions, then they must have other relations which satisfy other purely abstract conditions.
Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about. So far is this view of mathematics from being obvious, that we can easily assure ourselves that it is not, even now, generally understood. For example, it is habitually thought that the certainty of mathematics is a reason for the certainty of our geometrical knowledge of the space of the physical universe. This is a delusion which has vitiated much philosophy in the past, and some philosophy in the present.
(p. 28) So far as our observations are concerned, we are not quite accurate enough to be certain of the exact conditions regulating the things we come across in nature. But we can by a slight stretch of hypothesis identify these observed conditions with some one set of the purely abstract conditions. In doing so, we make a particular determination of the group of unspecified entities which are the relata in the abstract science. In the pure mathematics of geometrical relationships, we say that, if any group entities enjoy any relationships among its members satisfying this set of abstract geometrical conditions, then such-and-such additional abstract conditions must also hold for such relationships. But when we come to physical space, we say that some definitely observed group of physical entities enjoys some definitely observed relationships among its members which do satisfy this above-mentioned set of abstract geometrical conditions. We thence conclude that the additional relationships which we conclude to hold in any such case, must also therefore hold in this particular case.
To take another example from arithmetic. It is a general abstract truth of pure mathematics that any group of forty entities can be subdivided into two groups of twenty entities. We are therefore justified in concluding that a particular group of apples which we believe to contain forty members can be subdivided into two groups of apples of which each contains twenty members. But there always remains the possibility that we have miscounted the big group; so that, when we come in practice to subdivide it, we shall find that one of the two heaps has an apple too few or an apple too many.
(p. 29-30) In criticising an argument based upon the application of mathematics to particular matters of fact there are always three processes to be kept perfectly distinct in our minds.
(1) We must first scan the purely mathematical reasoning to make sure that there are no mere slips in it—no casual illogicalities due to mental failure.
(2) The next process is to make quite certain of all the abstract conditions which have been presupposed to hold. This is the determination of the abstract premises from which the mathematical reasoning proceeds. (2.1) The chief danger is that of oversight, namely, tacitly to introduce some condition, which it is naturally for us to presuppose, but which in fact need not always be holding. (2.2) There is another opposite oversight in this connection which does not lead to error, but only to lack of simplification.
(3) The third process of criticism is that of verifying that our abstract postulates hold for the particular case in question. It is in respect to his process of verification for the particular case that all the trouble arises. There are two distinct questions involved. (3.1) There are particular definite things observed, and we have to make sure that the relations between these things really do obey certain definite exact abstract conditions. (3.2) But another question arises. The things directly observed are, almost always, only samples. We want to conclude that the abstract conditions, which hold for the samples, also hold for all other entities which, for some reason or other, appear to us to be of the same sort. The process of reasoning from the sample to the whole species is Induction.
(p. 31) Whitehead draws a comparison between language and mathematics with regard to their abstractive power. Latin and Greek were inflected languages. This means that they express an unanalysed complex of ideas by the mere modification of a word. But in a language such as English, where propositions and auxiliary verbs are used to drag into the open the whole bundle of ideas involved, there is the overwhelming gain in explicitness. With this we can now see, says Whitehead, what is the function in thought which is performed by pure mathematics. It is a resolute attempt to go the whole way in the direction of complete analysis, so as to separate the elements of mere matter of fact from the purely abstract conditions which they exemplify.
With this we achieve, says Whitehead, first (1) direct aesthetic appreciation of the content of experience. That is an apprehension of what this experience is in itself in its own particular essence, including its immediate concrete values. (p. 32) (2) Then there is the abstraction of the particular entities involved, viewed in themselves, and as apart from the particular occasions of experience in which we are then apprehending them. (3) Lastly there is the further apprehension of the absolutely general conditions satisfied by the particular relations of those entities as in that experience.
There is nonetheless a limitation in mathematical generality; it is a qualification which applies equally to all general statements. No statement, except one, can be made respecting any remote occasion which enters into no relationship with the immediate occasion so as to form a constitutive element of the essence of that immediate occasion. This one statement is: if anything out of relationship, then complete ignorance as to it.
(p. 33) Either we know something of the remote occasion by the cognition which is itself an element of the immediate occasion or we know nothing. Accordingly, the full universe, disclosed for every variety of experience, is a universe in which every detail enters into its proper relationship with the immediate occasion.
The generality of mathematics is the most complete generality consistent with the community of occasions which constitutes our metaphysical situation.
It is further to be noticed that the particular entities require these general conditions for their ingression into any occasions; but the same general conditions may be required by many types of particular entities. This fact, that the general conditions transcend any one set of particular entities, is the ground for the entry into mathematics, and into mathematical logic, of the notion of the “variable”. It is by the employment of this notion that general conditions are investigated without any specification of particular entities.
In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key (k) to it.
This pattern of relationships among general abstract conditions is imposed alike on external reality, and on our abstract representations of it, by the general necessity that every thing must be just its own individual self, with its own individual way of differing from everything else.
(p. 34) This is nothing else than the necessity of abstract logic, which is the presupposition involved in the very fact of inter-related existences as disclosed in each immediate occasion of experience.
Whitehead now returns to the aforementioned concept of a key (k) to the pattern of general abstract conditions. From a select set (s) of those general conditions—in any one and the same occasion—a pattern involving an infinite variety of other such conditions—in the same occasion—can be developed by the pure exercise of abstract logic. Any such select set (s) is called the set of postulates (p), or premises, from which the reasoning (r) proceeds. The reasoning (r) is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates (p).
This exhibition of reason (r) [or the harmony of the logical reason, which divines the complete pattern as involved in the postulates] is the most aesthetic property arising from the mere fact of concurrent existence in the unity of one occasion.
Wherever there is (i) a unity of occasion there is thereby established an (ii) aesthetic relationship between the general conditions involved in that occasion. This aesthetic relationship is that which is divined in the exercise of rationality. Whatever falls within that relationship is thereby exemplified in that occasion, whatever falls without that relationship is thereby excluded from exemplification in that occasion.
The complete pattern of general conditions, thus exemplified, is determined by any one of many select sets (s) of these conditions. These key sets (s) are sets of equivalent postulates.
The reasonable harmony of being, which is required for the unity of a complex occasion, together with the completeness of the realisation (in that occasion) of all that is involved in its logical harmony, is the primary article of metaphysical doctrine. It means that for things to be together involves that they are reasonably together. (p. 35) This means that thought can penetrate into every occasion of fact, so that by comprehending its key conditions, the whole complex of its pattern of conditions lies open before it.
The logical harmony involved in the unity of an occasion is both exclusive and inclusive. The occasion must exclude the inharmonious, and it must include the harmonious.
Whitehead now turns his attention to the father of mathematics, Pythagoras. An important question that Pythagoras asked, according to Whitehead, was “What is the status of mathematical entities, such as numbers for example, in the realm of things?” The number two, for example, is in some sense exempt from the flux of time and the necessity of position in space. Yet it is involved in the real world.
(p. 36) It is also said that Pythagoras thought that the mathematical entities, such as numbers and shapes, were the ultimate stuff out of which the real entities of our perceptual experience are constructed.
The Platonic world of ideas is the refined, revised form of the Pythagorean doctrine that number lies at the base of the real world.
In a sense, Plato and Pythagoras stand nearer to modern physical science than does Aristotle. In support to this, Whitehead ads the following remark. (p. 37) The practical counsel to be derived from Pythagoras, is to measure, and thus to express quality in terms of numerically determined quantity. But the biological sciences, then and till our time, have been overwhelmingly classificatory. Accordingly, Aristotle by his Logic throws the emphasis on classification. The popularity of Aristotelian Logic retarded the advance of physical science throughout the Middle Ages. If only the schoolmen had measured instead of classifying, how much they might have learnt!
(p. 38) In the seventeenth century the influence of Aristotle was at its lowest, and mathematics recovered the importance of its earlier period. But the mathematics, which now emerged into prominence, was a very different science from the mathematics of the earlier epoch. It had gained in generality, and had started upon its almost incredible modern career of piling subtlety of generalisation upon subtlety of generalisation.
Algebra now came upon the science, and algebra is a generalisation of arithmetic. In the same way as the notion of number abstracted from reference to any one particular set of entities, so in algebra abstraction is made from the notion of any particular numbers.
(p. 40) The point which Whitehead is trying to make is that this dominance of the idea of functionality in the abstract sphere of mathematics found itself reflected in the order of nature under the guise of mathematically expressed laws of nature.
As a particular example of this Whitehead mentions the notion of periodicity (cf. 6). The general recurrences of things are very obvious in our ordinary experience. Days recur, lunar phases recur, the seasons of the year recur, rotating bodies recur to their old positions, beats of the heart recur, breathing recurs. Apart from recurrence, knowledge would be impossible; for nothing could be referred to our past experience. Also, apart from some regularity of experience, measurement would be impossible. In our experience, as we gain the idea of exactness, recurrence is fundamental.
The following are some of the major contributions to the idea of periodicity in the sixteenth and seventeenth century. (1) Kepler divined a law connecting the major axes of the planetary orbits with the periods in which the planets respectively described their orbits; (2) Galileo observed the periodic vibrations of pendulums; (3) Newton explained sound as being due to the disturbance of air by the passage through it of periodic waves of condensation and rarefaction; (4) Huyghens explained light as being due to the transverse waves of vibration of a subtle ether; (5) Mersenne connected the period of the vibration of a violin string with its density, tension, and length.
(p. 42) As the result of the prominence of mathematicians in the seventeenth century, the eighteenth century was mathematically minded, more especially where French influence predominated. An exception must be made of the English empiricism derived from Locke. Outside France, Newton’s direct influence on philosophy is best seen in Kant, and not in Hume.
In the nineteenth century, the general influence of mathematics waned. The romantic movement in literature, and the idealistic movement in philosophy were not the products of mathematical minds. But this does not mean that mathematics was being neglected, or even that it was uninfluential. During the nineteenth century pure mathematics made almost as much progress as during all the preceding centuries from Pythagoras onwards.
(p. 43) In truth, Whitehead says, there were only ever two periods of mathematical influence upon European thought. (1) The first period was that stretching from Pythagoras to Plato, when the possibility of the science, and its general character, first dawned upon the Grecian thinkers. (2) The second period comprised the seventeenth and eighteenth centuries of our modern epoch. Both periods had certain common characteristics. Whitehead has in mind a certain contrast between two opposing sides that drove the progress forward.
In the age of (1) Pythagoras, for example, (a) there were waves of religious enthusiasm, seeking direct enlightenment into the secret depths of being; (b) and at the opposite pole, there was the awakening of critical analytical thought, probing with cool dispassionateness into ultimate meanings.
(2) Regarding the second, modern epoch, the pagan mysteries may be compared to the Puritan reaction and to the Catholic reaction (critical scientific interest was alike in both epochs, though with minor differences of substantial importance).
(p. 44) Having said that, Whitehead nonetheless emphasises, that we should not press the parallels between the two epoch too far. In short, the modern world is far larger and more complex.
Whitehead now turns his attention to modern mathematics and physics.
In order to explain exactly how mathematics is gaining in general importance at the present time, let us start from a particular scientific perplexity.
(p. 45) At present physics is troubled by the quantum theory. One of the most hopeful lines of explanation is to assume that an electron does not continuously traverse its path in space. This alternative notion as to its mode of existence is that it appears at a series of discrete positions in space which it occupies for successive durations of time. It is as though an automobile moving at the average rate, did not traverse the road continuously; but appeared successively at the successive milestones, remaining for two minutes at each milestone.
First the question is purely mathematical. But later the problem is handed over to philosophers. This discontinuous existence in space, thus assigned to electrons, is very unlike the continuous existence of material entities which we habitually assume as obvious. The electron seems to be borrowing the character which some people have assigned to the Mahatmas of Tibet. Accordingly if this explanation is allowed, we have to revise all our notions of the ultimate character of material existence.
(p. 46) A steadily sounding note is explained as the outcome of vibrations in the air: a steady colour is explained as the outcome of vibrations in the ether. If we explain the steady endurance of matter on the same principle, we shall conceive each primordial element as a vibratory ebb and flow of an underlying energy, or activity.
(i) Each primordial element will be an organised system of vibratory streaming of energy; (ii) there will be a definite period associated with each element; (iii) within that period the stream-system will sway from one stationary maximum to another; (iv) the system forming the primordial element, is nothing at any instant it requires its whole period in which to manifest itself (like a note of music).
If we divide time into smaller elements, the vibratory system as one electronic entity has no existence.
(p. 47-48) Regarding evidence for such a theory, Whitehead posits the following. The whole theory centres round the radiant energy from an atom, and is intimately associated with the periods of the radiant wave-systems.
A new problem is now placed before philosophers and physicists, if we entertain the hypothesis that the ultimate elements of matter are in their essence vibratory. By this is meant that apart from being a periodic system, such an element would have no existence. With this hypothesis we have to ask, what are the ingredients which form the vibratory organism.
The field is now open for the introduction of some new doctrine of organism which may take the place of the materialism with which, since the seventeenth century, science has saddled philosophy.
