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      “the axiomatic foundation of physics cannot be derived from experience, but must be freely invented.”  Such invention must be guided by confidence “that nature represents the realization of the simplest that is mathematically thinkable,” and by the hope that through pure mathematical construction “pure thought is capable of comprehending reality”.

      “Quantum mechanics calls for a great deal of respect ....... The theory offers a lot, but it hardly brings us any closer to the Old Mans secret.  For my part, at least, I am convinced that he doesn’t throw dice.”

Albert Einstein

    ‘We proceeded straight from plane geometry to solid bodies in motion without considering solid bodies first on their own.  The right thing is to proceed from second dimension to third, which brings us, I suppose, to cubes and other three-dimensional figures.’

    ‘Thats true enough,’ he agreed, ‘but the subject is one which doesn’t seem to have been explored yet, Socrates.’

    ‘For two reasons,’ I replied. ‘There is no state which sets any value on it, and so, being difficult, it is not pursued with energy; and the pursuit is not likely to be successful without a director, who is difficult to find and, even if found, is unlikely to be obeyed in the present intolerant mood of those who study the subject.  But, under the general direction of a state that sets a value on it, their obedience would be assured, and investigations pressed forward continuously and energetically till the problems were cleared up.  Even now, with all the neglect and inadequate treatment it has suffered from the public and from students who do not understand its real uses, the subject is so attractive that it makes progress inspite of all handicaps, and it would not be surprising if a solution of its problems were to appear.’

    ‘Yes, it has very great attractions,’ he said.  ‘But explain more clearly what you said just now.  You said geometry dealt with plane surfaces.’

    ‘Yes.’

    ‘Then you first said astronomy came next, but subsequently went back on what you had said.’

    ‘More haste less speed,’ I said. ‘In my hurry I overlooked solid geometry, which should come next, because it’s so absurdly undeveloped, and put astronomy, which is concerned with solids in motion, after plane [and solid] geometry.’

    ‘Yes, that’s what you did,’ he agreed.

    ‘Then let us put astronomy fourth, and assume that the neglect of solid geometry would be made good under state encouragement.’

Plato - ‘The Republic’ (Education of the Philosopher  528 bcde).

    “We must in my opinion begin by distinguishing between that which always is and never becomes from that which is always becoming but never is.  The one is apprehensible by intelligence with the aid of reasoning, being eternally the same, the other is the object of opinion and irrational sensation, coming to be and ceasing to be, but never fully real.  In addition, everything that becomes or changes must do so owing to some cause; for nothing can come to be without a cause.  Whenever, therefore, the maker of anything keeps his eye on the eternally unchanging and uses it as his pattern for the form and function of his product the result must be good; whenever he looks to something that has come to be and uses a model that has come to be, the result is not good.

Plato - ‘Timaeus’ (28)

     “Mathematical ideas are more difficult for us to envisage than for Plato, for he was used to Pythagoreon theory which saw basic arithmetic notions as representative of something not only existent in the physical world but really a force within it.”

Plato - ‘Phaedo’ (postscript)

      “Among the studies of natural causes and laws, it is light that most delights it’s students.  Among all the great branches of mathematics, the certainty of it’s demonstrations pre-eminently elevates the minds of it’s investigators.  Perspective, therefore, should be preferred above all mans discourses and disciplines.  In this subject the visual rays are elucidated by means of demonstrations which derive their glory not only from mathematics but also from physics; the one is adorned equally with the flowers of the other.”

Leonardo da Vinci quoting John Pecham from the thirteenth century.

Martin Kemp - ‘The Science of Art’ (p5)

      “Artistic (and also scientific) creation is such mental activity as brings dimly perceived feelings (or thoughts) to such a degree of clearness that these feelings or thoughts are transmitted to other people”.

      “True science investigates and brings to human perception such truths and such knowledge as the people of a given time and society consider most important.  Art transmits these truths from the region of perception to the region of emotion.  If therefore the path chosen by science be false, so also will be the path taken by art.  Science gives direction to the forward movement; while art causes the actual progression.  Thus a false activity of science inevitably causes a correspondingly false activity of art.”

Leo Tolstoy - ‘What is Art’ (conclusion XX)

       “Of the chemistry of his day and generation, Kant declared that it was a science, but not Science - eine Wissenschaft, aber nicht Wissenschaft - for that the criterion of true science lay in it’s relation to mathematics.  This was an old story : for Roger Bacon had called mathematics porta et clavis scientiarum, and Leonardo da Vinci had said much the same.  Once again, a hundred years after Kant, Du Bois-Reymond, profound student of the many sciences on which physiology is based, recalled the old saying, and declared that chemistry would only reach the rank of science, in the high and strict sense, when it should be found possible to explain chemical reactions in the light of their causal relations to the velocities, tensions and conditions of equilibrium of the constituent molecules; that, in short, the chemistry of the future must deal with molecular mechanics by the methods and in the strict language of mathematics, as the astronomy of Newton and Laplace dealt with the stars in their courses.  We know how great a step was made towards this distant goal as Kant described it, when van’t Hoff laid the firm foundations of a mathematical chemistry, and earned his proud epitaph - Physicam chemiae adiunxit..

We need not wait for the full realisation of Kant’s desire, to apply to the natural sciences the principle which he laid down.  Though chemistry fall short of its ultimate goal in mathematical mechanics, nevertheless physiology is vastly strengthened and enlarged by making use of the chemistry, and of the physics, of the age.  Little by little it draws nearer to our conception of a true science with each branch of physical science which it brings into relation with itself: with every physical law and mathematical theorem which it learns to take into its employ.  Between the physiology of Haller, fine as it was, and that of Liebig, Helmholtz, Ludwig, Claude Bernard, there was all the difference in the world.

    As soon as we adventure on the paths of the physicist, we learn to weigh and to measure, to deal with time and space and mass and their related concepts, and to find more and more our knowledge expressed and our needs satisfied through the concept of number, as in the dreams and visions of Plato and Pythagoras; for modern chemistry would have gladdened the hearts of those great philosophic dreamers.  Dreams apart, numerical precision is the very soul of science, and its attainment affords the best, perhaps the only criterion of the truth of theories and correctness of experiments.  So said Sir John Herschel, a hundred years ago; and Kant had said that it was Nature herself, and not the mathematician, who brings mathematics into natural philosophy.

    But the zoologist or morphologist has been slow, where the physiologist has long been eager, to invoke the aid of the physical or mathematical sciences; and the reasons for this difference lie deep, and are partly rooted in old tradition and partly in the diverse minds and temperaments of men.  To treat the living body as a mechanism was repugnant, and seemed even ludicrous, to Pascal; and Goethe, lover of nature as he was, ruled mathematics out of place in natural history.  Even now the zoologist has scarce begun to dream of defining in mathematical language even the simplest organic forms.  When he meets with a simple geometrical construction, for instance in the honeycomb, he would fain refer it to psyshical, or to skill and ingenuity, rather than to the operation of physical forces or mathematical laws; When he sees in snail, or nautilus, or tiny foraminiferal or radiolarian shell a close approach to sphere or spiral, he is prone of old habit to believe that after all it is something  more than a spiral or a sphere, and that in this ‘something more’ there lies what neither mathematics nor physics can explain.  In short, he is deeply reluctant to compare the living with the dead, or to explain by geometry or by mechanics the things which have their part in the mystery of life.  Moreover he is little inclined to feel the need of such explanations, or of such extension of his field of thought.  ............”

D’arcy Thompson - ‘On Growth and Form’ (Introductory)

“Is seeing proving?  What we see is influenced by our cultural and educational conditioning.  Presuppositions are difficult to escape from and therefore it is equally impossible to describe what we see in objective language.”

“All scientific findings based on observation and induction must remain conjectures and temporary.  Induction can never offer you the certainty that logic can.”

“Any science based on induction will never be certain and will always have a problem with it’s human and questionable empirical base.”

David Hume

“ ........ the choice between competing paradigms regularly raises questions that cannot be resolved by the criteria of normal science.  To the extent, as significant as it is incomplete, that two scientific schools disagree about what is a problem and what a solution, they will inevitably talk through each other when debating the relative merits of their respective paradigms.  In the partially circular arguments that regularly result, each paradigm will be shown to satisfy more or less the criteria that it dictates for itself and to fall short of a few of those dictated by its opponent......... The normal scientific tradition that emerges from a scientific revolution is not only incompatible but often actually incommensurable with that which has gone before........ Because it has that character, the choice is not and cannot be determined merely by the evaluative procedures characteristic of normal science, for these depend in part upon a particular paradigm, and that paradigm is at issue.”

Darwin, in a particularly perceptive passage at the end of his Origin of Species, wrote: “Although I am fully convinced of the truth of the views given in this volume…… I by no means expect to convince experienced naturalists whose minds are stocked with a multitude of facts all viewed, during a long course of years, from a point of view directly opposite to mine……[B]ut I look with confidence to the future, - to young and rising naturalists, who will be able to view both sides of the question with impartiality.”  And Max Planck, surveying his own career in his Scientific Autobiography, sadly remarked that “a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it.”

Thomas Kuhn ‘The Structure of Scientific Revolutions’  Chapter IX & XII

The inertia of the human mind and its resistance to innovation are most clearly demonstrated not, as one might expect, by the ignorant mass – which is easily swayed once its imagination is caught – but by professionals with a vested interest in tradition and in the monopoly of learning.  Innovation is a twofold threat to academic mediocrities: it endangers their oracular authority, and it evokes the deeper fear that their whole, laboriously constructed intellectual edifice might collapse. The academic backwoodsmen have been the curse of genius from Aristarchus to Darwin and Freud; they stretch, a solid and hostile phalanx of pedantic mediocrities, across the centuries.

Arthur Koestler ‘Sleepwalkers‘  p 433

Just as a well trained pet will obey his master no matter how great the confusion in which he finds himself, and no matter how urgent the need to adopt new patterns of behaviour, so in the very same way a well-trained rationalist will obey the mental image of his master, he will conform to the standards of argumentation he has learned, he will adhere to these standards no matter how great the confusion in which he finds himself, and he will be quite incapable of realising that what he regards as the ‘voice of reason’ is but a causal after-effect of the training he had received. He will be quite unable to discover that the appeal to reason to which he succumbs so readily is nothing but a political manoeuvre. 

Paul Feyerabend ‘Against Method’ chapter One

For first, to speak generally, an argument from authority to wiser examinations is but a weaker kind of proof, it being but a topical probation, and as we term it, an inartificial [artless, clumsy, natural] argument, depending upon a naked asseveration; wherein neither declaring the causes, affections [properties] or adjuncts of what we believe, it carrieth not with it the reasonable inducements of knowledge. And therefore ‘Against [one] who denies first principles [it is useless to argue]’, ‘he said so himself’ [implying that the assertion is arbitrary and unsupported] or ‘the pupil ought to believe [what the teacher tells him]’, although postulates very accommodable unto junior indoctrinations, yet are their authorities but temporary, and not to be embraced beyond the minority of our intellectuals…

Thomas Browne - ‘Pseudodoxia Epidemica’ I.vi-vii.40-47

      “The charm of knowledge would be small if so much shame did not have to be overcome on the road to it”

Nietzsche - ‘Beyond good and evil’

      “Thus the happier branches of knowledge are those which are more nearly related to folly, and by far the happiest men are those who have no traffic at all with any kind of learning and follow nature for their only guide.”

Erasmus - ‘Praise of Folly’

It may strike us today as somewhat strange to see a genius like Leonardo describe perspective as the “reign and rudder of painting,” and to hear a powerfully imaginative artist like Paolo Ucello answer his wife’s request that he finally come to bed with the now-hackneyed phrase, “But how sweet perspective is!” All we can do is try to imagine what this achievement meant then. For not only did it elevate art to a “science” (and for the Renaissance that was an elevation): the subjective visual impression was indeed so far rationalised  that this very impression could itself become the foundation for a solidly grounded and yet, in an entirely modern sense, “infinite” experimental world. (One could even compare the function of Renaissance perspective with that of critical philosophy, and the function of Greco-Roman perspective with that of skepticism.) The result was a translation of psychophysiological space into mathematical space ; in other words, an objectification of the subjective.

Erwin Panofsky - ‘Perspective as Symbolic Form’ section III

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