TEMPERATURE AND PRESSURE

Everybody talks about the weather, and that usually means the temperature, an inescapable part of our environment. Yet Newton's laws of mechanics tell us nothing about temperature. Is there any connection between mechanics and temperature?

In Chapter 13 we saw a connection. If you drop a block from above a table, its potential energy first turns into kinetic energy, and then is transformed into thermal energy when the block hits the table. After a while the only evidence that those events occurred is a slight warming of the surroundings, that is, a small increase in temperature.

What really happens is that the kinetic energy of the falling block is turned into the energy of motion of atoms and molecules. The energy is still there, but the motions are in random directions, not the organized motion of a whole block of matter.

THE DEVELOPMENT OF DIFFERENTIAL CALCULUS

After the advent of algebra in the sixteenth century, mathematical discoveries inundated Europe. The most important were differential calculus and integral calculus, bold new methods for attacking a host of problems that had challenged the world's best minds for more than 2000 years. Differential calculus deals with ideas such as speed, rate of growth, tangent lines, and curvature, whereas integral calculus treats topics such as area, volume, arc length, and centroids.

Work begun by Archimedes in the third century B.C. led ultimately to the birth of integral calculus in the seventeenth century. This development has a long and fascinating history to which we shall return in Chapter 7.

Differential calculus has a relatively short history. Its principles were first formulated early in the seventeenth century when a French mathematician, Pierre de Fermat, tried to devise a way of finding the smallest and largest values of a given function. He imagined the graph of a function having, at each of its points, a direction given by a tangent line, as suggested by the points labeled in Fig. 3.1.

THE RISE OF VECTOR ANALYSIS

Throughout their history, mathematics and physics have been intimately related; a discovery in one field led to an improvement in the other. Early natural philosophers grappling with quantities such as distance, speed, and time used geometry inherited from the Greeks to explore physical problems. But in the tumultuous years of the seventeenth century, physics underwent a transformation – a shift in emphasis from numerical quantities, such as distance and speed, to vector quantities, such as displacement and velocity, which have direction as well as magnitude.

The transition was neither abrupt nor confined to that century. It was necessary to invent new mathematical objects – vectors – and new mathematical machinery for manipulating them – vector algebra – to embody the properties of the physical quantities they were to represent.

FORMS OF ENERGY

The concept of energy, as we saw in Chapter 13, is subtle, elegant, and rich. It describes a dynamic property of the universe which is strictly and absolutely conserved; energy can neither be created nor destroyed. Not even in the presence of friction is energy ever lost; it is simply transformed into other forms. Nevertheless, the universe is winding down. Energy tends to be transformed from well-organized forms into more disorganized forms, until it becomes completely useless.

In the First Meditation reasons are provided which give us possible grounds for doubt about all things, especially material things, so long as we have no foundations for the sciences other than those which we have had up till now. Although the usefulness of such extensive doubt is not apparent at first sight, its greatest benefit lies in freeing us from all our preconceived opinions, and providing the easiest route by which the mind may be led away from the senses. The eventual result of this doubt is to make it impossible for us to have any further doubts about what we subsequently discover to be true.

In the Second Meditation, the mind uses its own freedom and supposes the non-existence of all the things about whose existence it can have even the slightest doubt; and in so doing the mind notices that it is impossible that it should not itself exist during this time. This exercise is also of the greatest benefit, since it enables the mind to distinguish without difficulty what belongs to itself, i.e. to an intellectual nature, from what belongs to the body. But since some people may perhaps expect arguments for the immortality of the soul in this section, I think they should be warned here and now that I have tried not to put down anything which I could not precisely demonstrate.

Distinguished Sir,

In criticizing my Meditations you have produced a very elegant and careful essay which I think will be of great benefit in shedding light on their truth. So I consider that I am greatly indebted both to you, for writing down your objections, and to the Reverend Father Mersenne for encouraging you to do so. That gentleman is extremely anxious to inquire into all things, and is an indefatigable supporter of everything that furthers the glory of God; he very well knows that the best way of discovering whether my arguments deserve to be regarded as true demonstrations is to subject them to the examination and vigorous attack of a number of critics of outstanding learning and intelligence, so as to test whether I can give a satisfactory reply to all their objections. With this in mind he has challenged as many people as possible to attempt such a critique; and some, including, I am delighted to say, yourself, have agreed to his request. In fact you have not so much used philosophical arguments to refute my opinions as employed various debating skills to get round them; but this itself is a source of pleasure to me, since it leads me to suppose that it will not be easy to produce any further arguments against me, apart from those contained in the preceding objections of my other critics, which you have read.

Before the publication of the first edition of these Meditations, I wanted them to be examined not only by the Doctors of the Sorbonne, but also by all other learned men who would take the trouble to scrutinize them. My aim was to have their objections and my replies published as a continuation of the Meditations, following the order in which they had been produced, and I hoped that this would serve to make the truth more evident. The Fifth Set of Objections which were sent to me did not seem to me to be the most important, and they were extremely long; but nonetheless I agreed to have them published in their appropriate place out of courtesy to their author. I even allowed him to see the proofs, to prevent anything being printed of which he did not approve. But since that time he has produced a large volume containing his original objections together with several new ‘counter-objections’ or answers to my replies. In this book he complains of my publishing his objections, as if I had done so against his will, and says that he sent them to me only for my private instruction. Because of this, I am quite happy to oblige him now by removing his objections from the present volume, and this is why, when I learnt that M. Clerselier was taking the trouble to translate the other sets of objections, I asked him to omit the fifth set.

I will now shut my eyes, stop my ears, and withdraw all my senses. I will eliminate from my thoughts all images of bodily things, or rather, since this is hardly possible, I will regard all such images as vacuous, false and worthless. I will converse with myself and scrutinize myself more deeply; and in this way I will attempt to achieve, little by little, a more intimate knowledge of myself. I am a thing that thinks: that is, a thing that doubts, affirms, denies, understands a few things, is ignorant of many things, is willing, is unwilling, and also which imagines and has sensory perceptions; for as I have noted before, even though the objects of my sensory experience and imagination may have no existence outside me, nonetheless the modes of thinking which I refer to as cases of sensory perception and imagination, in so far as they are simply modes of thinking, do exist within me – of that I am certain.

In this brief list I have gone through everything I truly know, or at least everything I have so far discovered that I know. Now I will cast around more carefully to see whether there may be other things within me which I have not yet noticed. I am certain that I am a thinking thing. Do I not therefore also know what is required for my being certain about anything?

I briefly touched on the topics of God and the human mind in my Discourse on the method of rightly conducting reason and seeking the truth in the sciences, which was published in French in 1637. My purpose there was not to provide a full treatment, but merely to offer a sample, and learn from the views of my readers how I should handle these topics at a later date. The issues seemed to me of such great importance that I considered they ought to be dealt with more than once; and the route which I follow in explaining them is so untrodden and so remote from the normal way, that I thought it would not be helpful to give a full account of it in a book written in French and designed to be read by all and sundry, in case weaker intellects might believe that they ought to set out on the same path.

In the Discourse I asked anyone who found anything worth criticizing in what I had written to be kind enough to point it out to me. In the case of my remarks concerning God and the soul, only two objections worth mentioning were put to me, which I shall now briefly answer before embarking on a more precise elucidation of these topics.

It remains for me to examine whether material things exist. And at least I now know they are capable of existing, in so far as they are the subject-matter of pure mathematics, since I perceive them clearly and distinctly. For there is no doubt that God is capable of creating everything that I am capable of perceiving in this manner; and I have never judged that something could not be made by him except on the grounds that there would be a contradiction in my perceiving it distinctly. The conclusion that material things exist is also suggested by the faculty of imagination, which I am aware of using when I turn my mind to material things. For when I give more attentive consideration to what imagination is, it seems to be nothing else but an application of the cognitive faculty to a body which is intimately present to it, and which therefore exists.

To make this clear, I will first examine the difference between imagination and pure understanding. When I imagine a triangle, for example, I do not merely understand that it is a figure bounded by three lines, but at the same time I also see the three lines with my mind's eye as if they were present before me; and this is what I call imagining.

Sir,

The task of defending the Author of all things against a new race of giants, and of demonstrating his existence, is one which you have undertaken with such great success that from now on men of good will can hope that no one who carefully reads your Meditations will fail to acknowledge the existence of an eternal power on whom every single thing depends. We therefore wanted to draw your attention to various passages, which are indicated below, and ask you to clarify them, so that, as far as possible, there may be nothing left in your work which is not clearly demonstrated. You have trained your mind by continual meditations for several years, so that what seems doubtful and very obscure to others is quite clear to you; indeed, you may have a clear mental intuition of these matters and perceive them as the primary and principal objects of the natural light. We are simply pointing out the issues on which it seems worthwhile to burden you with the task of providing a clearer and more extended explanation and demonstration. You have embarked on your arguments for the greater glory of God and the immense benefit of mankind and, once you have done what we ask, there will scarcely be anyone left who can deny that they do indeed have the force of demonstrations.

[Bourdin]

You ask me many questions, distinguished Sir, regarding your new method for investigating the truth, and you beg – indeed you insistently demand – that I should give my reply. But I shall keep silent and refuse to humour you unless you agree to the following. First, let us throughout our discussion completely ignore the contributions of those who have written or spoken on this subject. Further, will you please frame your questions in such a way as to avoid appearing to ask about the views of others, or their intentions, or the results of their work, or whether or not their opinions were correct. I ask you to behave as if no one had ever had any views or written or spoken anything on these matters, and to ask only the questions which seem to you to present some difficulty as you meditate and pursue your new method of philosophizing. This will enable us to search for the truth and to do so in a way which will keep safe and intact those laws of friendship and respect which should govern the dealings of learned men. Since you signify your agreement, and promise to follow this suggestion, I shall respond appropriately. And so we may proceed.

Gentlemen,

You have indeed called up a mighty opponent to challenge me, and his intelligence and learning could well have caused me serious difficulty had he not been a good and kind theologian who preferred to befriend the cause of God, and its humble champion, rather than to mount a serious attack. But though it was extremely kind of him to pull his punches, it would not be so acceptable for me to keep up the pretence; and hence I would rather expose his carefully disguised assistance to me than answer him as if he were an adversary.

First of all he summarizes my chief argument for proving the existence of God, thus helping to fix it all the more firmly in the reader's memory. And after briefly conceding the claims which he considers to have been demonstrated with sufficient clarity, thereby adding the weight of his own authority to them, he raises the one question which gives rise to the most important difficulty, namely the question of what should be understood by the term ‘idea’ in this context, and of whether such an idea requires a cause of any sort.

Now I wrote that an idea is the thing which is thought of in so far as it has objective being in the intellect. But to give me an opportunity of explaining these words more clearly the objector pretends to understand them in quite a different way from that in which I used them.

Gentlemen,

I read with great pleasure the comments which you made on my little book dealing with First Philosophy. They make me appreciate both your goodwill towards me and your piety towards God and zeal to further his glory. And I cannot but be very happy, not only because you have thought my arguments worthy of examination, but also because I think I can give you a reasonably adequate reply to all the criticisms that you make.

First, you warn me to remember that my rejection of the images of bodies as delusive was not something I actually and really carried through, but was merely a fiction of the mind, enabling me to draw the conclusion that I was a thinking thing; and I should not suppose that it followed from this that I was in fact nothing more than a mind. I But I already showed that I was quite well aware of this in the Second Meditation, where I said ‘Yet may it not perhaps be the case that these very things which I am supposing to be nothing, because they are unknown to me, are in reality identical with the “I” of which I am aware? I do not know, and for the moment I shall not argue the point.’ Here I wanted to give the reader an express warning that at that stage I was not yet asking whether the mind is distinct from the body, but was merely examining those of its properties of which I can have certain and evident knowledge.

As soon as he had completed the Meditations, Descartes began to circulate them among his friends, asking for comments and criticisms. He also sent the manuscript to Friar Marin Mersenne (1588–1648), his friend and principal correspondent, asking him to obtain further criticisms. He wrote to Mersenne in a letter of 28 January 1641: ‘I shall be glad if people make me as many objections as possible – and the strongest ones they can find. For I hope that in consequence the truth will stand out all the better.’ The resulting six sets of Objections (the first set collected by Descartes himself, the remainder by Mersenne) were published in Latin, together with Descartes' Replies, in the same volume as the first (1641) edition of the Meditations. The second edition of the Meditations (1642) contained in addition the Seventh Set of Objections together with Descartes' Replies, and also the Letter to Dinet (all in Latin). The terms ‘Objections’ and ‘Replies’ were suggested by Descartes himself, who asked that his own comments should be called ‘Replies’ rather than ‘Solutions’ in order to leave the reader to judge whether his replies contained solutions to the difficulties offered (letter to Mersenne, 18 March 1641).

The volume containing the French translation of the Meditations (by de Luynes), which appeared in 1647, also contained a French version of the first six sets of Objections and Replies by Descartes' disciple Claude Clerselier (1614–84).

During these past few days I have accustomed myself to leading my mind away from the senses; and I have taken careful note of the fact that there is very little about corporeal things that is truly perceived, whereas much more is known about the human mind, and still more about God. The result is that I now have no difficulty in turning my mind away from imaginable things and towards things which are objects of the intellect alone and are totally separate from matter. And indeed the idea I have of the human mind, in so far as it is a thinking thing, which is not extended in length, breadth or height and has no other bodily characteristics, is much more distinct than the idea of any corporeal thing. And when I consider the fact that I have doubts, or that I am a thing that is incomplete and dependent, then there arises in me a clear and distinct idea of a being who is independent and complete, that is, an idea of God. And from the mere fact that there is such an idea within me, or that I who possess this idea exist, I clearly infer that God also exists, and that every single moment of my entire existence depends on him. So clear is this conclusion that I am confident that the human intellect cannot know anything that is more evident or more certain.