A Quote by Pierre-Simon Laplace
It is interesting thus to follow the intellectual truths of analysis in the phenomena of nature. This correspondence, of which the system of the world will offer us numerous examples, makes one of the greatest charms attached to mathematical speculations.
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In the last two months I have been very busy with my own mathematical speculations, which have cost me much time, without my having reached my original goal. Again and again I was enticed by the frequently interesting prospects from one direction to the other, sometimes even by will-o'-the-wisps, as is not rare in mathematic speculations.
Our present work sets forth mathematical principles of philosophy. For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces. It is to these ends that the general propositions in books 1 and 2 are directed, while in book 3 our explanation of the system of the world illustrates these propositions.
Our design, not respecting arts, but philosophy, and our subject, not manual, but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore we offer this work as mathematical principles of philosophy; for all the difficulty of philosophy seems to consist in this from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena.
There are several kinds of truths, and it is customary to place in the first order mathematical truths, which are, however, only truths of definition. These definitions rest upon simple, but abstract, suppositions, and all truths in this category are only constructed, but abstract, consequences of these definitions ... Physical truths, to the contrary, are in no way arbitrary, and do not depend on us.
I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts... When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through the process of 'seeing'.
Poetry is related to philosophy as experience is related to empirical science. Experience makes us acquainted with the phenomenon in the particular and by means of examples, science embraces the whole of phenomena by means of general conceptions. So poetry seeks to make us acquainted with the Platonic Ideas through the particular and by means of examples. Philosophy aims at teaching, as a whole and in general, the inner nature of things which expresses itself in these. One sees even here that poetry bears more the character of youth, philosophy that of old age.
Prayer is not a way to get what we want to happen, like the remote control that comes with the television set. I think that prayer may be less about asking for the things we are attached to than it is about relinquishing our attachments in some way. It can take us beyond fear, which is an attachment, and beyond hope, which is another form of attachment. It can help us remember the nature of the world and the nature of life, not on an intellectual level but in a deep and experiential way. When we pray, we don't change the world, we change ourselves. We change our consciousness.
This harmony that human intelligence believes it discovers in nature - does it exist apart from that intelligence? No, without doubt, a reality completely independent of the spirit which conceives it, sees it or feels it, is an impossibility. A world so exterior as that, even if it existed, would be forever inaccessible to us. But what we call objective reality is, in the last analysis, that which is common to several thinking beings, and could be common to all; this common part, we will see, can be nothing but the harmony expressed by mathematical laws.
Mathematical analysis is as extensive as nature itself; it defines all perceptible relations, measures times, spaces, forces, temperatures:;; this difficult science is formed slowly, but it preserves every principle which it has once acquired; it grows and strengthens itself incessantly in the midst of the many variations and errors of the human mind. It's chief attribute is clearness; it has no marks to express confused notations. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them.
The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work-that is, correctly to describe phenomena from a reasonably wide area.
...learning chiefly in mathematical sciences can so swallow up and fix one's thought, as to possess it entirely for some time; but when that amusement is over, nature will return, and be where it was, being rather diverted than overcome by such speculations.
Whether we live by the seaside, or by the lakes and rivers, or on the prarie, it concerns us to attend to the nature of fishes, since they are not phenomena confined to certain localities only, but forms and phases of the life in nature universally dispersed. The countless shoals which annually coast the shores of Europe and America are not so interesting to the student of nature as the more fertile law itselffrom which it results that they may be found in water in so many places, in greater or lesser numbers.
Nothing in all Nature is more certain than the fact that no single thing or event can stand alone. It is attached to all that has gone before it, and it will remain attached to all that will follow it. It was born of some cause, and so it must be followed by some effect in an endless chain.
Mitt Romney will tell us the hard truths we need to hear to end the debacle of putting the world's greatest care system in the hands of federal bureaucrats and putting those bureaucrats between an American citizen and her doctor.
If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.
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