An explanation of the historical development of English mathematics and higher education by Norbert Wiener, in an obituary for G. H. Hardy – with some parallels for today

Once you read enough books you begin to come across curious bits of knowledge that are rarely mentioned nowadays, often in the most obscure sources.

Although George Harold Hardy (1877-1947) was very well known in mathematics circles in the early to mid 20th century, he never attained the prominence even then of other intellectual peers of his generation, such as Bertrand Russell, Albert Einstein, etc., Nowadays he is a remote figure known only to passionate students of the mathematics and the history thereof.

However, thanks to a very illuminating obituary written in memorial by Norbert Wiener in 1947, a fascinating view of the historical development of mathematics, and more broadly that of the scientific and higher education system in England from the 17th to 20th century, comes to light.

Here are the key passages:

“…

Hardy came from a family with artistic and intellectual traditions. He went to Winchester and then to Trinity College, Cambridge. The milieu in which he developed as a mathematician is one which it is particularly difficult for those outside of the English tradition to understand, and even rather difficult for those belonging to the newer English tradition which Hardy himself had so much hand in establishing.

It all goes back to the disputes between Newton and Leibniz concerning the invention of the calculus. At present we have not much doubt of the fact that Newton invented the differential and integral calculus, that Leibniz’ work was somewhat later but independent, and that Leibniz’ notation was far superior to Newton’s. At the beginning the relations between the Leibnizian and the Newtonian schools were not hostile, but it was not long before patriotic and misguidedly loyal colleagues of both discoverers instigated a quarrel, the effects of which have scarcely yet died out. the British mathematicians to use the less flexible Newtonian notation and to affect to look down on the new work done by the Leibnizian school on the Continent. For a while there was no scarcity of able English mathematicians of the strictly Newtonian school. For example, we must mention Taylor and Maclaurin. However, when the great continental school of the Bernoullis and Euler arose (not to mention Lagrange and Laplace who came later) there were no men of comparable calibre north of the Channel to compete with them on anything like a plane of equality.

Part of this must be attributed to the fallen status the British mathematicians to use the less flexible Newtonian notation and to affect to look down on the new work done by the Leibnizian school on the Continent. For a while there was no scarcity of able English mathematicians of the strictly Newtonian school. For example, we must mention Taylor and Maclaurin. However, when the great continental school of the Bernoullis and Euler arose (not to mention Lagrange and Laplace who came later) there were no men of comparable calibre north of the Channel to compete with them on anything like a plane of equality. Part of this must be attributed to the fallen status of the English Universities during the 18th century.

In the 17th century the English Universities were seats of learning comparable with the greatest schools of the Continent, but in the 18th century the grasping new Whig aristocracy that had risen out of the prosperous middle class (the nabobs) took over the older English institutions, the common land, public schools, universities, lock, stock and barrel, as their private property. The public schools were transformed from institutions of a semi-charitable nature to the place where the children of the new aristocracy were formed after its own pattern. The universities became nests of sinecures for dependent clergymen. In this atmosphere creative scholarship did not and could not flourish, and it is not until the 19th century is well under way that we find the signs of a new awareness of what the continental scholars, particularly Laplace and Lagrange, had done in mathematics. Among the English names belonging to this tentative reformation we may mention Boole, Peacock and DeMorgan. DeMorgan in particular is associated with the new University College at London which by its pressure did so much to bring the older universities back to a sense of intellectual responsibility.

This reform of English education was far from complete. The level of mathematics at Oxford was for many years scarcely more than contemptible, and even at Cambridge the training was devoted to the passing of severe examinations, the Triposes, rather than to the development of original mathematical workers. What mathematical talent there was in the British Isles went rather to the formation of a great school of mathematical physicists. Even here Cambridge entered the game rather late. Clerk Maxwell owes more to Faraday, the self-taught practical experimentor, than to any Cambridge man, and neither George Green nor Hamilton was in the Cambridge tradition. Sylvester, as a Jew, was not permitted to enter the older universities till towards the end of his life, and is another of those seminal figures who center around the University of London. Cayley is the first real great Cambridge pure mathematician of the 19th century. He certainly was in touch with those continental scholars whose interest was primarily in algebra, but algebra was at that time an important secondary mathematical subject rather than one in the main stream of development.

It is not remarkable that in such an environment, secluded from the central activity of world mathematics, mathematical study should be devoted rather to the formation of public school ushers or a trial intellectual run for promising barristers than to research activities. As a matter of fact, the Tripos was made such an ordeal, at least in difficulty though in general not in originality, that it marked the culminating point in the intellectual life of many of those who participated in it, and their subsequent activity became retrospective rather than creative. This was the state of English mathematics to about the turn of the century, when an awareness of the great work of the continental mathematicians smuggles itself into England by non-academic bypaths. The English generation of pure mathematicians of the 19th century and the first decade of the 20th century is curiously tentative. It has many important names, such as A. N. Whitehead, Andrew Forsyth, E. A. Hobson and W. H. Young. These all carry to some degree a mathematical style and ethos formed under the older English tradition into a period when the topics of interest were far more continental.

In addition to his accomplishments in research and teaching, Hardy contributed greatly to the reform of mathematical instruction. He was bitterly opposed to the rigid and unmathematical Tripos system and is unquestionably in a large part responsible for the fact that the order of rank of the Wranglers, those who obtain first class in the Tripos, has not been published since 1912. The present mathematical Tripos and indeed the whole system of training at Cambridge has been modified in the sense of conforming very closely to the actual work and career of the mathematicians of this day. Even this change, which has spread from Cambridge to all the British Universities, is a compromise between the old system and a system where research should even more completely take the place of examinations.

…”

Curious indeed when contemplating alongside with the current trends!

From Volume IV of Norbert Wiener: Collected Works

Samuel Johnson’s fascinating epistolary writing, ‘The Rambler No. 42’

Note that this is likely a fictional letter written by himself instead of an actual reader, from https://www.johnsonessays.com/the-rambler/misery-modish-solitude/

Originally published August 11, 1750, revised in 1756

A later editor added the title ‘The misery of a modish lady in solitude’


Mihi tarda fluunt ingrataque tempora.

Horace. liber primus I. Epistle 1. 15.

How heavily my time revolves along.

Elphinston

TO THE RAMBLER.

MR. RAMBLER,

I am no great admirer of grave writings, and therefore very frequently lay your papers aside before I have read them through; yet I cannot but confess that, by slow degrees, you have raised my opinion of your understanding, and that, though I believe it will be long before I can be prevailed upon to regard you with much kindness, you have, however, more of my esteem than those whom I sometimes make happy with opportunities to fill my tea-pot, or pick up my fan. I shall therefore chuse you for the confidant of my distresses, and ask your counsel with regard to the means of conquering or escaping them, though I never expect from you any of that softness and pliancy, which constitutes the perfection of a companion for the ladies: as, in the place where I now am, I have recourse to the mastiff for protection, though I have no intention of making him a lap-dog.

My mamma is a very fine lady, who has more numerous and more frequent assemblies at her house than any other person in the same quarter of the town. I was bred from my earliest infancy in a perpetual tumult of pleasure, and remember to have heard of little else than messages, visits, playhouses, and balls; of the awkwardness of one woman, and the coquetry of another; the charming convenience of some rising fashion, the difficulty of playing a new game, the incidents of a masquerade, and the dresses of a court-night. I knew before I was ten years old all the rules of paying and receiving visits, and to how much civility every one of my acquaintance was entitled; and was able to return, with the proper degree of reserve or of vivacity, the stated and established answer to every compliment; so that I was very soon celebrated as a wit and a beauty, and had heard before I was thirteen all that is ever said to a young lady. My mother was generous to so uncommon a degree as to be pleased with my advance into life, and allowed me, without envy or reproof, to enjoy the same happiness with herself; though most women about her own age were very angry to see young girls so forward, and many fine gentlemen told her how cruel it was to throw new chains upon mankind, and to tyrannize over them at the same time with her own charms, and those of her daughter.

I have now lived two-and-twenty years, and have passed of each year nine months in town, and three at Richmond; so that my time has been spent uniformly in the same company, and the same amusements, except as fashion has introduced new diversions, or the revolutions of the gay world have afforded new successions of wits and beaux. However, my mother is so good an economist of pleasure, that I have no spare hours upon my hands; for every morning brings some new appointment, and every night is hurried away by the necessity of making our appearance at different places, and of being with one lady at the opera, and with another at the card-table.

When the time came of settling our schemes of felicity for the summer, it was determined that I should pay a visit to a rich aunt in a remote county. As you know the chief conversation of all tea-tables, in the spring, arises from a communication of the manner in which time is to be passed till winter, it was a great relief to the barrenness of our topicks, to relate the pleasures that were in store for me, to describe my uncle’s seat, with the park and gardens, the charming walks and beautiful waterfalls; and every one told me how much she envied me, and what satisfaction she had once enjoyed in a situation of the same kind.

As we are all credulous in our own favour, and willing to imagine some latent satisfaction in any thing which we have not experienced, I will confess to you, without restraint, that I had suffered my head to be filled with expectations of some nameless pleasure in a rural life, and that I hoped for the happy hour that should set me free from noise, and flutter, and ceremony, dismiss me to the peaceful shade, and lull me in content and tranquillity. To solace myself under the misery of delay, I sometimes heard a studious lady of my acquaintance read pastorals, I was delighted with scarce any talk but of leaving the town, and never went to bed without dreaming of groves, and meadows, and frisking lambs.

At length I had all my clothes in a trunk, and saw the coach at the door; I sprung in with ecstasy, quarrelled with my maid for being too long in taking leave of the other servants, and rejoiced as the ground grew less which lay between me and the completion of my wishes. A few days brought me to a large old house, encompassed on three sides with woody hills, and looking from the front on a gentle river, the sight of which renewed all my expectations of pleasure, and gave me some regret for having lived so long without the enjoyment which these delightful scenes were now to afford me. My aunt came out to receive me, but in a dress so far removed from the present fashion, that I could scarcely look upon her without laughter, which would have been no kind requital for the trouble which she had taken to make herself fine against my arrival. The night and the next morning were driven along with inquiries about our family; my aunt then explained our pedigree, and told me stories of my great grandfather’s bravery in the civil wars, nor was it less than three days before I could persuade her to leave me to myself.

At last economy prevailed; she went in the usual manner about her own affairs, and I was at liberty to range in the wilderness, and sit by the cascade. The novelty of the objects about me pleased me for a while, but after a few days they were new no longer, and I soon began to perceive that the country was not my element; that shades, and flowers, and lawns, and waters, had very soon exhausted all their power of pleasing, and that I had not in myself any fund of satisfaction, with which I could supply the loss of my customary amusements.

I unhappily told my aunt, in the first warmth of our embraces, that I had leave to stay with her ten weeks. Six only yet are gone, and how shall I live through the remaining four? I go out and return; I pluck a flower, and throw it away; I catch an insect, and when I have examined its colours set it at liberty; I fling a pebble into the water, and see one circle spread after another. When it chances to rain, I walk in the great hall, and watch the minute-hand upon the dial, or play with a litter of kittens, which the cat happens to have brought in a lucky time.

My aunt is afraid I shall grow melancholy, and therefore encourages the neighbouring gentry to visit us. They came at first with great eagerness to see the fine lady from London; but when we met, we had no common topick on which we could converse; they had no curiosity after plays, operas, or musick: and I find as little satisfaction from their accounts of the quarrels or alliances of families, whose names, when once I can escape, I shall never hear. The women have now seen me, know how my gown is made, and are satisfied; the men are generally afraid of me, and say little, because they think themselves not at liberty to talk rudely.

Thus I am condemned to solitude; the day moves slowly forward, and I see the dawn with uneasiness, because I consider that night is at a great distance. I have tried to sleep by a brook, but find its murmurs ineffectual; so that I am forced to be awake at least twelve hours, without visits, without cards, without laughter, and without flattery. I walk because I am disgusted with sitting still, and sit down because I am weary with walking. I have no motive to action, nor any object of love, or hate, or fear, or inclination. I cannot dress with spirit, for I have neither rival nor admirer. I cannot dance without a partner; nor be kind or cruel, without a lover.

Such is the life of Euphelia; and such it is likely to continue for a month to come. I have not yet declared against existence, nor called upon the destinies to cut my thread; but I have sincerely resolved not to condemn myself to such another summer, nor too hastily to flatter myself with happiness. Yet I have heard, Mr. Rambler, of those who never thought themselves so much at ease as in solitude, and cannot but suspect it to be some way or other my own fault, that, without great pain, either of mind or body, I am thus weary of myself: that the current of youth stagnates, and that I am languishing in a dead calm, for want of some external impulse. I shall therefore think you a benefactor to our sex, if you will teach me the art of living alone; for I am confident that a thousand and a thousand ladies, who affect to talk with ecstasies of the pleasures of the country, are in reality, like me, longing for the winter, and wishing to be delivered from themselves by company and diversion.

I am, Sir, Yours,

Euphelia.


Due to the changing fashions over time even the greatest writers certainly no longer write like this anymore!

If you are likewise fascinated after reading I would highly recommend reading his other essays, freely accessible online. Note that it’s likely most, if not all, his ‘reader letters’ were actually fictional letters.

Nonetheless still impressive 270 years later.

A further analysis of Johnson’s epistles can be found here, JSTOR Arts & Sciences VII, paywalled.

Plato’s caves all the way up! — no ultimate limits to complexity?

Many have heard of the allegory of Plato’s cave, of the wonderment, or more likely bewilderment, that the tortured souls experience when they first leave the cave and see the ‘actual world’ versus the mere projected shadows they had subsisted on.

There is an interesting unstated assumption, with corollary possibility, here. 

That there is only one level of cave, that what is ‘outside’ of the cave are not the projections of an even greater cave. 

i.e. the cave dwellers, who originally took the shadows at face value, realize their true nature with the help of a new frame of reference. Yet this does not automatically confer on them the capability to competently assess this more complex environment which after all could be higher level shadows!

To make the assumption of a single cave layer, if you’ll bear with me, seems like dimensional arrogance. Assuming that what is outside Plato’s cave constitutes ‘reality’ makes about as much sense from the perspective of the fourth spatial dimension as to assume that the cave shadows constitutes ‘reality’. In both cases they are indistinguishable from, and likely are, the projections of higher dimensional ‘reality’.

This leads to the possibility of an infinite number of ever greater, dimensionally nested, caves. As there are no mathematical or logical reasons to limit how many there could be.

Which leads to a conundrum doesn’t it? 

After all we universally accept without question the existence of 2D projections from a 3D object, and mathematicians and physicists likewise for 3D projections. There shouldn’t be an arbitrary limit on this right?

The following idea is potentially so enormous that it is difficult to even put to words.

If it’s possible for the ‘outside’ to be the shadows of a greater cave, how do we know we are not projected shadows? 

Furthermore, it’s understood that a single 3D object can take on many different 2D forms on Plato’s cave wall. To infer the object from any given cave shadow can be highly misleading. 

So how do we know what objects are at all? 

In fact, given arbitrary volume and projection energy, it seems likely a sufficiently complex 3D object can produce all possible lower dimensional shadows. (As a corollary I wonder if there is, or could be, a proof for this.)

Given these heady ideas it’s comforting to note, if true, the absence of any ultimate cutoffs. 

Possibilities are truly boundless!

evolution of trust

Here’s a truly amazing educational web game that concisely explains the fundamental determinants needed for trust to evolve. Highly recommend checking it out.

Broadly speaking, the environment and the people within co-evolve, this suggest that the appropriate environment is a necessary condition for high trust behaviour to come about. Perhaps there is a way to facilitate the organization and positive feedback loops of trust building over the net? A TrustNet so to speak.

Who was the first human?

Consider this thought experiment:

1. You would consider your parents as human right?

2. Their parents were human too right? i.e. your grandparents

3. Your grandparent’s parent’s were also human right?

4. And their’s, and their’s, and their’s, ….

5. Well if you keep on doing this for thousands upon thousands generations eventually we are in the year 5 million BCE. …

And wait there were no ‘humans’ this long ago!

So whence did people start becoming ‘human’?